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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/| T ENGiNEEftiNG UBRAfT c,3 i^ale 'Btcentenmal ^uhlvcatm^ ELEMENTARY PRINCIPLES IN STATISTICAL MECHANICS I^ale 'Bicentennial l^ublication^ With the approval of the President and Fellows of Tale University^ a series of volumes has been prepared by a number of the Professors and In- structors^ to be issued in connection with the Bicentennial Anniversary^ as a partial indica- tion of the character of the studies in which the University teachers are engaged. This series of volumes is respectfully dedicated to ELEMENTARY PRINCIPLES IN STATISTICAL MECHANICS DEVELOPED WITH ESPECIAL REFERENCE TO THE RATIONAL FOUNDATION OF THERMODYNAMICS BY J. WILLARD GIBBS Professor of MathemaHcal Physics in Yale University NEW YORK : CHARLES SCRIBNER'S SONS LONDON: EDWARD ARNOLD 1902 Copyright, 190B, By Yale University Publishtd^ March, igo9 UNIVERSITY PRESS • JOHN WILSON AND SON • CAMBRIDGE, U.S.A. PREFACE. The usual point of view in the study of mechanic is that where the attention is mainly directed to the changes which take place in the course of time in a given system. The prin- cipal problem is the determination of the condition of the system with respect to configuration and velocities at any required time, when its condition in these respects has been given for some one time, and the fundamental equations are those which express the changes continuaUy taking place in the system. Inquiries of this kind are often simplified by taking into consideration conditions of the system otiier tiian those through which it actually passes or is supposed to pass, but our attention is not usually carried beyond conditions differing infinitesimally from those which are regarded as actual. For some purposes, however, it is desirable to take a broader view of the subject. We may imagine a great number of systems of the same nature, but differing in the configura- tions and velocities which they have at a given instant, and differing not merely infinitesimally, but it may be so as to embrace every conceivable combination of configuration and velocities. And here we may set the problem, not to follow a particular system through its succession of configurations, but to determine how the whole number of systems will be distributed among the various conceivable configurations and velocities at any required time, when the distribution has been given for some one time. The fundamental equation for this inquiry is that which gives the rate of change of the number of systems which fall within any infinitesimal limits of configuration and velocity. viii PREFACE. Such inquiries have been called by Maxwell statistical. They belong to a branch of mechanics which owes its origin to the desire to explain the laws of thermodynamics on mechan- ical principles, and of which Clausius, Maxwell, and Boltz- mann are to be regarded as the principal founders. The first inquiries in this field were indeed somewhat narrower in their scope than that which has been mentioned, being applied to the particles of a system, rather than to independent systems. Statistical inquiries were next directed to the phases (or con- ditions with respect to configuration and velocity) which succeed one another in a given system in the course of time. The explicit consideration of a great number of systems and their distribution in phase, and of the permanence or alteration of this distribution in the course of time is perhaps first found in Boltzmann's paper on the " Zusammenhang zwischen den Satzen fiber das Verhalten mehratomiger GasmolektQe mit Jacobi's Princip des letzten Multiplicators " (1871). But although, as a matter of histoiy, statistical mechanics owes its origin to investigations in thermodynamics, it seems eminently worthy of an independent development, both on account of the elegance and simplicity of its principles, and because it yields new results and places old truths in a new light in departments (laite outside of thermodynamics. More- over, the separate study of this branch of mechanics seems to afford the best foundation for the study of rational thermody- namics and molecular mechanics. The laws of thermodynamics, as empiricaUy determined, express the approximate and probable behavior of systems of a great number of particles, or, more precisely, they express the laws of mechanics for such systems as they appear to beings who have not the fineness of perception to enable them to appreciate quantities of the order of magnitude of those which relate to single particles, and who cannot repeat their experiments often enough to obtain any but the most probable results. The laws of statistical mechanics apply to conservative systems of any number of degrees of freedom, preface: ix and are exact. This does not make them more difficult to establish than the approximate laws for systems of a great many degrees of freedom, or for limited classes of such systems. The reverse is rather the case, for our attention is not diverted from what is essential by the peculiarities of the system considered, and we are not obliged to satisfy ourselves that the effect of the quantities and circumstances neglected will be negligible in the result. The laws of thermodynamics may be easily obtained from the principles of statistical me- chanics, of which they are the incomplete expression, but they make a somewhat blind guide in our search for those laws. This is perhaps the principal cause of the slow progress of rational thermodynamics, as contrasted with the rapid de- duction of the consequences of its laws as empirically estab- lished. To this must be added that the rational foundation of thermodynamics lay in a branch of mechanics of which the fundamental notions and principles, and the characteristic operations, were alike unfamiliar to students of mechanics. We may therefore confidently believe that nothing will more conduce to the clear apprehension of the relation of thermodynamics to rational mechanics, and to the interpreta- tion of observed phenomena with reference to their evidence respecting the molecular constitution of bodies, than the study of the fundamental notions and principles of that de- partment of mechanics to which thermodynamics is especially related. Moreover, we avoid the gravest difficulties when, giving up the attempt to frame hypotheses concerning the constitution of material bodies, we pursue statistical inquiries as a branch of rational mechanics. In the present state of science, it seems hardly possible to frame a dynamic theory of molecular action which shall embrace the phenomena of thermody- namics, of radiation, and of the electrical manifestations which accompany the union of atoms. Yet any theory is obviously inadequate which does not take account of all these phenomena. Even if we confine our attention to the X PREFACE. phenomena distinotively thermodynamic, we do not escape difficulties in as simple a matter as the number of degrees of freedom of a diatomic gas. It is well known that while theory would assign to the gas six degrees of freedom per molecule, in our experiments on specific heat we cannot ac- count for more than five. Certainly, one is building on an insecure foundation, who rests his work on hypotheses con- cerning the constitution of matter. Difficulties of this kind have deterred the author from at- tempting to explain the mysteries of nature, and have forced him to he contented with the more modest aim of deducing some of the more obvious propositions relating to the statis- tical branch of mechanics. Here, there can be no mistake in regard to the agreement of the hypotheses with the facts of nature, for nothing is assumed in that respect. The only error into which one can fall, is the want of agreement be- tween the premises and the conclusions, and this, with care, one may hope, in the main, to avoid. The matter of the present volume consists in large measure of results which have been obtained by the investigators mentioned above, although the point of view and the arrange- ment may be different. These results, given to the public one by one in the order of their discovery, have necessarily, in their original presentation, not been arranged in the most logical manner. In the first chapter we consider the general problem which has been mentioned, and find what may be called the funda- mental equation of statistical mechanics. A particular case of this equation will give the condition of statistical equi- librium, i. e., the condition which the distribution of the systems in phase must satisfy in order that the distribution shall be permanent. In the general case, the fundamental equation admits an integration, which gives a principle which may be variously expressed, according to the point of view from which it is regarded, as the conservation of density-in- phase, or of extension-in-phase, or of probability of phase. PREFACE. xi In the second chapter, we apply this principle of conserva- tion of probability of phase to the theory of errors in the calculated phases of a system, when the determination of the arbitrary constants of the integral equations are subject to error. In this application, we do not go beyond the usual approximations. In other words, we combine the principle of conservation of probability of phase, which is exact, with those approximate relations, which it is customary to assume in the " theory of errors." In the third chapter we apply the principle of conservation of extension-in-phase to the integration of the differential equations of motion. This gives Jacobi's " last multiplier," as has been shown by Boltzmann. In the fourth and following chapters we return to the con- sideration of statistical equilibrium, and confine our attention to conservative systems. We consider especially ensembles of systems in which the index (or logarithm) of probability of phase is a linear function of the energy. This distribution, on account of its unique importance in the theory of statisti- cal equilibrium, I have ventured to call canonical^ and the divisor of the energy, the moduluB of distribution. The moduli of ensembles have properties analogous to temperature, in that equality of the moduli is a condition of equilibrium with respect to exchange of energy, when such exchange is made possible. We find a differential equation relating to average values in the ensemble which is identical in form with the funda- mental differential equation of thermodynamics, the average index of probability of phase, with change of sign, correspond- ing to entropy, and the modulus to temperature. For the average square of the anomalies of the energy, we find an expression which vanishes in comparison with the square of the average energy, when the number of degrees of freedom is indefinitely increased. An ensemble of systems in which the number of degrees of freedom is of the same order of magnitude as the number of molecules in the bodies xii PREFACE. with which we experiment, if distributed canonically, would therefore appear to human observation as an ensemble of systems in which aU have the same energy. We meet with other quantities, in the development of the subject, which, when the number of degrees of freedom is very great, coincide sensibly with the modulus, and with the average index of probability, taken negatively, in a canonical ensemble, and which, therefore, may also be regarded as cor- responding to temperature and entropy. The correspondence is however imperfect, when the number of degrees of freedom is not very great, and there is nothing to recommend these quantities except that in definition they may be regarded as more simple than those which have been mentioned. In Chapter XIV, this subject of thermodynamic analogies is discussed somewhat at. length. Finally, in Chapter XV, we consider the modification of the preceding results which is necessary when we consider systems composed of a number of entirely similar -particles, or, it may be, of a number of particles of several kinds, all of each kind being entirely similar to each other, and when one of the variations to be considered is that of the numbers of the particles of the various kinds which are contained in a system. This supposition would naturally have been intro- duced earlier, if our object had been simply the expression of the laws of nature. It seemed desirable, however, to separate sharply the purely thermodynamic laws from those special modifications which belong rather to the theory of the prop- erties of matter. J. W. G. Nbw ILlybn, December, 1901. CONTENTS. CHAPTER I. GENERAL NOTIONS. THE PRINCIPLE OF CONSERVATION OF EXTENSION-IN-PHASE. Paob Hamilton's equations of motion 3-5 Ensemble of systems distributed in phase 5 Extension-in-phase, density-in-phase 6 Fundamental equation of statistical mechanics 6-8 Condition of statistical equilibrium 8 Principle of conservation of density-in-phase 9 Principle of conservation of extension-in-phase 10 Analogy in hydrodynamics 11 Extension-in-phase is an invariant 11-13 Dimensions of extension-in-phase 13 Various analytical expressions of the principle 13-15 Coefficient and index of probability of phase 16 Principle of conservation of probability of phase 17, 18 Dimensions of coefficient of probability of phase 19 CHAPTER n. APPLICATION OF THE PRINCIPLE OF CONSERVATION OP EXTENSION-IN-PHASE TO THE THEORY OF ERRORS. Approximate expression for the index of probability of phase . 20, 21 Application of the principle of conservation of probability of phase to the constants of this expression 21-25 CHAPTER ni. APPLICATION OF THE PRINCIPLE OF CONSERVATION OF EXTENSION-IN-PHASE TO THE INTEGRATION OF THE DIFFERENTIAL EQUATIONS OF MOTION. Case in which the forces are function of the coordinates alone . 26-29 Case in which the forces are functions of the coordinates with the time 30, 31 xiv CONTENTS. CHAPTER IV. ON THE DISTRIBUTION-IN-PHASB CALLED CANONICAL, IN WHICH THE INDEX OF PROBABILITY IS A LINEAR FUNCTION OF THE ENERGY. Paob Condition of statistical equilibrium 82 Other conditions which the coefficient of probability must satisfy . 83 Canonical distribution — Modulus of distribution 84 ^ must be finite 85 The modulus of the canonical distribution has properties analogous to temperature 85-87 Other distributions have similar properties 87 Distribution in which the index of probability is a linear function of the energy and of the moments of momentum about three axes . 88, 89 Case in which the forces are linear functions of the displacements, and the index is a linear function of the separate energies relating to the normal types of motion 89-41 Differential equation relating to average values in a canonical ensemble 42-44 This is identical in form with the fundamental differential equation of thermodynamics 44, 45 CHAPTER V. AVERAGE VALUES IN A CANONICAL ENSEMBLE OF SYS- TEMS. Case of V material points. Average value of kinetic energy of a single point for a given configuration or for the whole ensemble = |e . . 46,47 Average value of total kinetic energy for any given configuration or for the whole ensemble = f i^ 6 47 System of n degrees of freedom. Average value of kinetic energy, for any given configuration or for the whole ensemble =: § O . 48-50 Second proof of the same proposition 50-52 Distribution of canonical ensemble in configuration 52-54 Ensembles canonically distributed in configuration 55 Ensembles canonically distributed in velocity 56 CHAPTER VI. EXTEN8I0NI-IN-C0NFIGXJRATI0N AND EXTENSION-IN- VELOCITY. Extension-in-configuration and extension-in-velocity are invari- ants 57-59 CONTENTS. Paob Dimensions of these quantities 60 Index and coefficient of probability of configuration 61 Index and coefficient of probability of yelocitj 62 Dimensions of these coefficients 68 Relation between extension-in-configuration and extension-in-velocity 64 Definitions of extension-in-phase, extension-in-configuration, and ex- tension-in>yelocity, without explicit mention of coordinates • • 65-67 CHAPTER Vn. FARTHER DISCUSSION OF AVERAGES IN A CANONICAL ENSEMBLE OF SYSTEMS. Second and third differential equations relating to average values in a canonical ensemble 68, 69 These are identical in form with thermodynamic equations enun- ciated by Clausius 69 Average square of the anomaly of the energy — of the kinetic en- ergy — of the potential energy 70-72 These anomalies are insensible to human observation and experi- ence when the number of degrees of freedom of the system is very great 73, 74 Average values of powers of the energies 75-77 Average values of powers of the anomalies of the energies . . 77-80 Average values relating to forces exerted on external bodies . . 80-88 General formulae relating to averages in a canonical ensemble . 88-86 CHAPTER Vin. ON CERTAIN IMPORTANT FUNCTIONS OF THE ENERGIES OF A SYSTEM. Definitions. V = extension-in-phase below a limiting energy (e). ^-logdF/dt 87,88 Vq =5 extension-in-configuration below a limiting value of the poten- tial energy («,). 4>q^\ogdVqldMq 89»90 Vp = extension-in-velocity below a limiting value of the kinetic energy M' P^HdV^ld€p 90,91 Evaluation of Vp and <^ 91-93 Average values of functions of the kinetic energy 94, 95 Calculation of F from F^ 95,96 Approximate formulae for large values of n 97,98 Calculation of F or ^ for whole system when given for parts ... 98 (jeometrical illustration 99 xvi CONTENTS. CHAPTER ES:. THE FUNCTION ^ AND THE CANONICAL DISTRIBUTION. Paob When n > 2, the most probable value of the energy in a canonical ensemble is determined by (/<^/(/e — 1 /e 100, 101 When n > 2, the average value of d^ / c/e in a canonical ensemble isl/e 101 When n is large, the value of <^ corresponding to (/<^/(/e=l/0 ((^) is nearly equivalent (except for an additive constant) to the average index of probability taken negatively (— i;) • • 101-104 Approximate formulae for ^ + ^ when n is large 104-106 When n is large, the distribution of a canonical ensemble in energy follows approximately the law of errors 105 This is not peculiar to the canonical distribution 107, 108 Averages in a canonical ensemble 108-114 CHAPTER X. ON A DISTRIBUTION IN PHASE CALLED MICROCANONI- CAL IN WHICH ALL THE SYSTEMS HAVE THE SAME ENERGY. The microcanonical distribution defined as the limiting distribution obtained by various processes 115, 116 Average values in the microcanonical ensemble of functions of the kinetic and potential energies 117-120 If two quantities have the same average values in every microcanon- ical ensemble, they have the same average value in every canon- ical ensemble 120 Average values in the microcanonical ensemble of functions of the energies of parts of the system 121-123 Average values of functions of the kinetic energy of a part of the system 128, 124 Average values of the external forces in a microcanonical ensemble. Differential equation relating to these averages, having the form of the fundamental differential equation of thermodynamics . 124-128 CHAPTER XI. MAXIMUM AND MINIMUM PROPERTIES OP VARIOUS DIS- TRIBUTIONS IN PHASE. Theorems I-YI. Minimum properties of certain distributions . 129-138 Theorem YII. The average index of the whole system compared with the sum of the average indices of the parts 133-135 CONTENTS. xvii Paob Theorem VllL The average index of the whole ensemble com- pared with the average indices of parts of the ensemble • . 135-137 Theorem IX. Effect on the average index of making the distribu- tion-in-phase uniform within anj limits 137-138 CHAPTER Xn. ON THE MOTION OP SYSTEMS AND ENSEMBLES OF SYS- TEMS THBOUGH LONG PERIODS OF TIME. Under what conditions, and with what limitations, may we assume that a system will return in the course of time to its original phase, at least to any required degree of approximation? . • 139-142 Tendency in an ensemble of isolated systems toward a state of sta- tistical equilibrium 143-151 CHAPTER Xm. EFFECT OF VARIOUS PROCESSES ON AN ENSEMBLE OF SYSTEMS. Variation of the external coordinates can only cause a decrease in the average index of probability 152-154 This decrease may in general be diminished by diminishing the rapidity of the change in the external coordinates .... 154-157 The mutual action of two ensembles can only diminish the sum of their average indices of probability 158, 159 In the mutual action of two ensembles which are canonically dis- tributed, that which has the greater modulus will lose energy . 160 Repeated action between any ensemble and others which are canon- ically distributed with the same modulus will tend to distribute the first-mentioned ensemble canonically with the same modulus 161 Process analogous to a Carnot's cycle 162, 163 Analogous processes in thermodynamics 163, 164 CHAPTER XIV. DISCUSSION OF THERMODYNAMIC ANALOGIES. The finding in rational mechanics an h priori foundation for thermo- dynamics requires mechanical definitions of temperature and entropy. Conditions which the quantities thus defined must satisfy 165-167 The modulus of a canonical ensemble (e), and the average index of probability taken negatively (^), as analogues of temperature and entropy 167-169 xviii CONTENTS. Paob The functions of the energy dt/d log V and log V as analogoes of temperature and entropy 169-173 The functions of the energy dt/dtf) and <^ as analogues of tempera- ture and entropy 172-178 Merits of the different systems 178-183 If a system of a great number of degrees of freedom is microcanon- ically distributed in phase, any very small part of it may be re- garded as canonically distributed 188 Units of 6 and if compared with those of temperature and entropy 183-186 CHAPTER XV. SYSTEMS COMPOSED OF MOLECULES. Generic and specific definitions of a phase 187-189 Statistical equilibrium with respect to phases generically defined and with respect to phases specifically defined 189 Grand ensembles, petit ensembles 189, 190 Grand ensemble canonically distributed 190-193 Q must be finite 193 Equilibrium with respect to gain or loss of molecules .... 194-197 Average value of any quantity in grand ensemble canonically dis- tributed 198 Differential equation identical in form with fundamental differen- tial equation in thermodynamics 199, 200 Average value of number of any kind of molecules (y) . • . . 201 Average value of (y — y)^ 201,202 Comparison of indices 203-206 When the number of particles in a system is to be treated as variable, the average index of probability for phases generically defined corresponds to entropy 206 ELEMENTARY PRINCIPLES IN STATISTICAL MECHANICS ELEMENTARY PRINCIPLES IN STATISTICAL MECHANICS CHAPTER I. GENERAL NOTIONS. THE PRINCIPLE OF CONSERVATION OF EXTENSION-IN-PHASE. Wb shall use Hamilton's form of the equations of motion for a system of n degrees of freedom, writing g^ , . . .g„ for the (generalized) cotlrdinates, gi , . . . 2» for the (generalized) ve- locities, and for the moment of the forces. We shall call the quantities i\ , . . . -P„ the (generalized) forces, and the quantities p^ •••Pn't defined by the equations Pi = -j^f i^2 = -4, etc., (2) aqi aq2 where e^ denotes the kinetic energy of the system, the (gen- eralized) momenta. The kinetic energy is here regarded as a function of the velocities and coordinates. We shall usually regard it as a function of the momenta and coordinates,* and on this account we denote it by e^. This will not pre- vent us from occasionally using formulae like (2), where it is sufficiently evident the kinetic energy is regarded as function of the j's and g's. But in expressions like dcp/dq^^ , where the denominator does not determine the question, the kinetic * The use of the momenta instead of the yelocities as independent variables is the characteristic of Hamilton's method which gives his equations of motion their remarkable degree of simplicity. We shall find that the fundamental notions of statistical mechanics are most easily defined, and are expressed in the most simple form, when the momenta with the coordinates are used to describe the state of a system. ELEMENTARY PEINCIPLES IN STATISTICAL MECHANICS CHAPTER I. GENERAL NOTIONS. THE PRINCIPLE OF CONSERVATION OF EXTEN8I0N-IN-PHASE. Wb shall use Hamilton's form of the equations of motion for a system of n degrees of freedom, writing gj , . . .g„ for the (generalized) coordinates, gi, . . . 2» for the (generalized) ve- locities, and Fidq^ + F^dq^ ... + F,dq, (1) for the moment of the forces. We shall call the quantities i\ , . . . JJj the (generalized) forces, and the quantities p^ •••Pny defined by the equations aqi dq2 where e^ denotes the kinetic energy of the system, the (gen- eralized) momenta. The kinetic energy is here regarded as a function of the velocities and coordinates. We shall usually regard it as a function of the momenta and coordinates,* and on this account we denote it by ep. This will not pre- vent us from occasionally using formulae like (2), where it is sufficiently evident the kinetic energy is regarded as function of the j's and g's. But in expressions like dcp/dq^ , where the denominator does not determine the question, the kinetic * The use of the momenta instead of the velocities as independent yariables is the characteristic of Hamilton's method which gives his equations of motion their remarkable degree of simplicity. We shall find that the fundamental notions of statistical mechanics are most easily defined, and are expressed in the most simple form, when the momenta with the coordinates are used to describe the state of a system. 4 HAMILTON'S EQUATIONS. energy is always to be treated in the differentiation as function of the ^'s and ^'s. We have then These equations will hold for any forces whatever. If the forces are conservative, in other words, if the expression (1) is an exact differential, we may set where €^ is a function of the coordinates which we shall call the potential energy of the system. If we write e for the total energy, we shall have € = €p + e„ (5) and equations (3) may be written • de • de The potential energy (e^) may depend on other variables beside the coordinates Ji • . . ?». We shall often suppose it to depend in part on coordinates of external bodies, which we shall denote by a^j a^t etc. We shall then have for the com- plete value of the differential of the potential energy * c^e, = — J\ rfi7i . . — F^ dq^ -^ Aidoi — A^ c^Oa — etc., (7) where A^^ A^^ etc., represent forces (in the generalized sense) exerted by the system on external bodies. For the total energy (e) we shall have d€= ^i dpi . • • + g'n ^Pn — Pi ^^1 • • • — Pn <^2n — Ai doi — Ai da^ — etc. (8) It will be observed that the kinetic energy (e^) in the most general case is a quadratic function of the jp's (or j's) * It wiU be observed, that although we caU c the potential energy of the system which we are considering, it is really so defined as to include that energy which might be described as mutual to that system and external bodies. ENSEMBLE OF SYSTEMS. 6 involving also the y's but not the a*s ; that the potential energy, when it exists, is function of the j's and a's ; and that the total energy, when it exists, is function of the /?'s (or y's), the ^'s, and the a's. In expressions like dejdq^^ them's, and not the 9's, are to be taken as independent variables, as has already bJafted ^fl. »,pect to tokine* «n»gy. , Let us imagine a great number of independent systems, identical in nature, but differing in phase, that is, in their condition with respect to configuration and velocity. The forces are supposed to be determined for every system by the same law, being functions of the coordinates of the system ji, . . . jft, either alone or with the coordinates a^^ a^, etc. of certain external bodies. It is not necessary that they should be derivable from a force-function. The external coordinates a^, ag, etc. may vary with the time, but at any given time have fixed values. In this they differ from the internal coordinates jj, . . . j», which at the same time have different values in the different systems considered. Let us especially consider the number of systems which at a given instant faU within specified limits of phase, viz., those for which Pi ®^'» '^hich represent the increments of 2?i, Jx» ®^'» in the time dt shall be infinitely small in comparison with the infinitesimal differences p{' — p(^ q(' — j/, etc., which de- termine the magnitude of the element of extension-in-phase. The systems for which p^ passes the limit p( in the interval dt are those for which at the commencement of this interval the value of p^ lies between p^' and p^ — p^ dt, as is evident if we consider separately the cases in which p^^ is positive and negative. Those systems for which p^^ lies between these limits, and the other jt?'s and j's between the limits specified in (9), will therefore pass into or out of the element considered according as p is positive or negative, unless indeed they also pass some other limit specified in (9) during the same inter- val of time. But the number which pass any two of these limits will be represented by an expression containing the square oi dt as a, factor, and is evidently negligible, when dt is sufficiently small, compared with the number which we are seeking to evaluate, and which (with neglect of terms contain- ing dfi) may be found by substituting p^ dt for p/' — jt?/ in (10) or for rfpi in (11). The expression Dpi dt dpi. . . dp^ dqi . . . dq^ (13) will therefore represent, according as it is positive or negative, the increase or decrease of the number of systems within the given limits which is due to systems passing the limit p{. A similar expression, in which however D and p will have slightly different values (being determined for p/' instead of p(\ -vnll represent the decrease or increase of the number of systems due to the passing of the limit p^'. The difference of the two expressions, or dpi dpi . . . dp^ dqx . . . dq^ dt (14) 8 CONSERVATION OF will represent algebraically the decrease of the number of systems within the limits due to systems passing the limits p^ and jt?i". The decrease in the number of systems within the limits due to systems passing the limits j/ and y/' may be found in the same way. This will give (^(|p) + ^(^ (15) for the decrease due to passing the four limits p^^ p{\ 3/, q{'. But since the equations of motion (3) give ^ + ^ = 0, (16) the expression reduces to (y-Pi + j-'ii)^Ih'" dpn dq^... dq^ dt. (17) If we prefix 2 to denote summation relative to the suffixes 1 ... n, we get the total decrease in the number of systems within the limits in the time dt That is, S (j-Pi + -^qijdpi...dp^dqi...dq^dt=z — dD dpi . . . dpn dqi . • . dq^y (18) /dD\ ^ (dD . , dD ^\ where the suffix applied to the differential coefficient indicates that the p\ and y's are to be regarded as constant in the differ- entiation. The condition of statistical equilibrium is therefore If at any instant this condition is fulfilled for all values of the p's and j's, {dDldt)p^q vanishes, and therefore the condition will continue to hold, and the distribution in phase will be permanent, so long as the external co(5rdinates remain constant. But the statistical equilibrium would in general be disturbed by a change in the values of the external coordinates, which or DENSITY-IN'PHASE. 9 would alter the values of the jJ's as determined by equations (8)9 and thus disturb the relation expressed in the last equation. If we write equation (19) in the form ''* + 2(gi'i'^' + g?i'^*)=0. (21) /dD* ,. . dD p. 9 it will be seen to express a theorem of remarkable simplicity. Since D is a function of f, p^^ . . . j9„, Ji , . . • j«, its complete differential will consist of parts due to the variations of all these quantities. Now the first term of the equation repre- sents the increment of -D due to an increment of t (with con- • stant values of them's and j's), and the rest of the first member represents the increments of D due to increments of the jt?'s and y's, expressed by pi dt^ q^ dt^ etc. But these are precisely the increments which the j[?'s and j's receive in the movement of a system in the time dt The whole expression represents the total increment of D for the varying phase of a moving system. We have therefore the theorem : — In an ensemble of mechanical systems identical in nature and subject to forces determined by identical lawSj but distributed in phase in any continuous manner j the density-inrphase is constant in time for the varying phases of a moving system; provided^ that the forces of a system are functions of its co- ordinates^ either alone or with the time* This may be called the principle of conservation of density- in-phase. It may also be written where a, ... A represent the arbitrary constants of the integral equations of motion, and are suflSxed to the differential co- * The condition that the forces Fi,.,.Fn are functions of ^i > • • • 9» and 01,03, etc., which last are functions of the time, is analytically equivalent to the condition that Fi, ...Fn are functions of qi, ...gn and the time. Explicit mention of the external coordinates, 01,03, etc., has been made in the preceding pages, because our purpose will require us hereafter to con- sider these coordinates and the connected forces, ^1, A^, etc., which repre- sent the action of the systems on external bodies. 10 CONSERVATION OF efficient to indicate that they are to be regarded as constant in the differentiation. We may give to this principle a slightly different expres- sion. Let us call the value of the integral I ... I dpi , • . dpn dqi , • • dq^ (23) taken within any limits the extension-in-phase within those lunits. When the phases bounding an extension-in-phase vary in the course of time according to the dynamical laws of a system svibject to forces which are functions of the coordinates either alone or with the time^ the value of the extension-in-phase thus hounded remains constant. In this form the principle may be called the principle of conservation of extermonAnrphase. In some respects this may be regarded as the most simple state- ment of the principle, since it contains no explicit reference to an ensemble of systems. Since any extension-in-phase may be divided into infinitesi- mal portions, it is only necessary to prove the principle for an infinitely small extension. The number of systems of an ensemble which fall within the extension will be represented by the integral I ... I 2) dpx . • • dpj^ dqi . • . dqJ^• If the extension is infinitely small, we may regard D as con- stant in the extension and write 2) I ... I dpi . . . dpf^ dqi . • . dq^ for the number of systems. The value of this expression must be constant in time, since no systems are supposed to be created or destroyed, and none can pass the limits, because the motion of the limits is identical with that of the systems. But we have seen that D is constant in time, and therefore the integral I • • . I dpi . • . dp^ dqi • • • dq^^j EXTENSION'IN-PHASE. 11 which we have called the extension-in-phase, is also constant in time.* Since the system of coordinates employed in the foregoing discussion is entirely arbitrary, the values of the cotirdinates relating to any configuration and its immediate vicinity do not impose any restriction upon the values relating to other configurations. The fact that the quantity which we have called density-in-phase is constant in time for any given sys- tem, implies therefore that its value is independent of the coordinates which are used in its evaluation. For let the density-in-phase as evaluated for the same time and phase by one system of coordinates be -Di', and by another system Dg'. A system which at that time has that phase will at another time have another phase. Let the density as calculated for this second time and phase by a third system of coordinates be Dg"* Now we may imagine a system of coordinates which at and near the first configuration will coincide with the first system of coordinates, and at and near the second configuration •vnll coincide with the third system of coordinates. This will give Dj' = Dg". Again we may imagine a system of coordi- nates which at and near the first configuration will coincide with the second system of coordinates, and at and near the * If we regard a phase as represented by a point in space of 2 n dimen- sions, the changes which take place in the course of time in our ensemble of systems wiU be represented by a current in such space. This current will be steady so long as the external coordinates are not varied. In any case the current will satisfy a law which in its various expressions is analogous to the hydrodynamic law which may be expressed by the phrases conserva- tion of volumes or conservation of density about a moving point, or by the equation dx dy dz The analogue in statistical mechanics of this equation, viz., # + ^ + # + ^ + etc. = 0, ^Pl "9^1 "Pfl "^2 may be derived directly from equations (3) or (6), and may suggest such theorems as have been enunciated, if indeed it is not regarded as making them intuitively evident. The somewhat lengthy demonstrations given above will at least serve to give precision to the notions involved, and familiarity with their use. 12 EXTENSION'IN^PHASE second configuration will coincide with the third system of coordinates. This will give D^ = Dg". We have therefore A' = J>^'. It follows, or it may be proved in the same way, that the value of an extension-in-phase is independent of the system of coerdinates which is used in ite ^nation. This may easily be verified directly. If Ji? • . . ?», ^i, • •• Qn ^tre two systems of cotJrdinates, and |>x, . . . jt>„, P^, . . . P^ the cor- responding momenta, we have to prove that / ... I €lpi...dp^dqi,..dq^= I "' j ^^i" '^^ndQi>..dQny (24) when the multiple integrals are taken within limits consisting of the same phases. And this will be evident from the prin- ciple on which we change the variables in a multiple integral, if we prove that d{2>i9 • • •i?»>2'i> . • • 2n) where the first member of the equation represents a Jacobian or functional determinant. Since all its elements of the form dQ/dp are equal to zero, the determinant reduces to a product of two, and we have to prove that d(Pu...P,)d(Q,,...Q,) ^^ We may transform any element of the first of these deter- minants as follows. By equations (2) and (8), and in view of the fact that the ^'s are linear functions of the g's and therefore of the jt?'s, with coefficients involving the y's, so that a differential coefficient of the form dQ^/dp^ is function of the j's alone, we get * * The form of the equation d d€p d dtp dpy d(Xt dCU dp, in (27) reminds us of the fundamental identity in the differential calculus relating to the order of differentiation with respect to independent variables. But it will be observed that here the variables Qx and p, are not independent and that the proof depends on the linear relation between the Q's and the p*B, IS AN INVARIANT. 18 dP^g _ d^ d€j^ _/^( d^p d0^\ _ dPif dp, dQ^ r=i Wr dQ^ dp,, ) "" d ^/d€p dQ^\ _ _d^ d€p _ dq^ dQ^ f^i\d(ir dpj "" d(i^ dp^ "■ d(i. (27) But Since ^-K^.^^)' Therefore, dqy _dq, dQ. dQ, (28) (29) d(Pij ...Fn) _ d(qi, .. .qn) _ d(qi, * . . gn) d(Pu...Pn) d{Qu...) d{pT!', . . . qjf) dWy • • • 2'n) d(pi", . . . qn") d{pT!y . . . qj) ' where the double accents distinguish the values of the momenta and coordinates for a time if\ If we vary t, while if and if' remain constant, we have J^ d(pu " > g>) __ d(pi'f, . . . qjf) d^ d(pu . > » gn) .n4\ dtd{p^f,...qj)^ d(p^f,...qj) dtd{p^f\...qj^y ^ ^ Now since the time if' is entirely arbitrary, nothing prevents us from making if' identical with t at the moment considered. Then the determinant d{pi , > ' « gn) ^(M . . . ?.") will have unity for each of the elements on the principal diagonal, and zero for all the other elements. Since every term of the determinant except the product of the elements on the principal diagonal will have two zero factors, the differen- tial of the determinant will reduce to that of the product of these elements, i. «., to the sum of the differentials of these elements. This gives the equation d d(pi j...qn) _ dpi t^PjLi.^ , dqn Now since t = ^', the double accents in the second member of this equation may evidently be neglected. This will give, in virtue of such relations as (16), EXTENSION'IN-PHASE. 16 which substituted in (34) will give d d(pi , ...qn) _Q dt d(pi'y . . . qj) The determinant in this equation is therefore a constant, the value of which may be determined at the instant when t=zfy when it is evidently unity. Equation (38) is therefore demonstrated. Again, if we write a, ... A for a system of 2 w arbitrary con- stants of the integral equations of motion, p^^ j^, etc. wiU be functions of a, ... A, and ^, and we may express an extension- in-phase in the form If we suppose the limits specified by values of a, • • . A, a system initially at the limits will remain at the limits. The principle of conservation of extension-in-phase requires that an extension thus bounded shaU have a constant value. This requires that the determinant under the integral sign shall be constant, which may be written t /^"'-1'^ = 0. (36) This equation, which may be regarded as expressing the prin- ciple of conservation of extension-in-phase, may be derived directly from the identity d(a, ... A) d(pi'y . . . qj) d{a, ... A) in connection with equation (33). Since the coordinates and momenta are functions of a, ... A, and tj the determinant in (36) must be a function of the same variables, and since it does not vary with the tune, it must be a function of a, ... A alone. We have therefore ^^1;^ = fun. (a,... A). (37) 16 CONSERVATION OF It is the relative numbers of systems which fall within dif- ferent limits, rather than the absolute numbers, with which we are most concerned. It is indeed only with regard to relative numbers that such discussions as the preceding will apply with literal precision, since the nature of our reasoning implies that the number of systems in the smallest element of space which we consider is very great. This is evidently inconsist- ent with a finite value of the total number of systems, or of the density-in-phase. Now if the value of D is infinite, we cannot speak of any definite number of systems within any finite limits, since all such numbers are infinite. But the ratios of these infinite numbers may be perfectiy definite. If we write iVfor the total number of systems, and set ^ = §. (38) P may remain finite, when N and D become infinite. The integral Pdpi...dq^ (39) /•••/■ taken within any given limits, will evidentiy express the ratio of the number of systems falling within those limits to the whole number of systems. This is the same thing as the probability that an unspecified system of the ensemble (i. e. one of which we only know that it belongs to the ensemble) will lie within the given limits. The product Pdpi...dq^ (40) expresses the probability that an unspecified system of the ensemble will be found in the element of extension-in-phase dpx . . . dq^. We shall call P the coefficient of probability of the phase considered. Its natural logarithm we shall call the index of probability of the phase, and denote it by the letter ?;. If we substitute NP and Ne^ for 2> in equation (19), we get (f)„=-^(S--^S*')- («) PROBABILITY OF PHASE. 17 The condition of statistical equilibrium may be expressed by equating to zero the second member of either of these equations. The same substitutions in (22) give and (§i....=»- <"> That is, the values of P and 17, like those of 2), are constant in time for moving systems of the ensemble. From this point of view, the principle which otherwise regarded has been called the principle of conservation of density-in-phase or conservation of extension-in-phase, may be called the prin- ciple of conservation of the coefficient (or index) of proba- bility of a phase varying according to dynamical laws, or more briefly, the principle of conservation of probability of phase. It is subject to the limitation that the forces must be functions of the coordinates of the system either alone or with the time. The application of this principle is not limited to cases in which there is a formal and explicit reference to an ensemble of systems. Yet the conception of such an ensemble may serve to give precision to notions of probability. It is in fact cus- tomary in the discussion of probabilities to describe anything which is imperfectly known as something taken at random from a great number of things which are completely described. But if we prefer to avoid any reference to an ensemble of systems, we may observe that the probability that the phase of a system falls within certain limits at a certain time, is equal to the probability that at some other time the phase will fall within the limits formed by phases corresponding to the first. For either occurrence necessitates the other. That is, if we write P' for the coefficient of probability of the phase pii • • • Qn at the time f , and P" for that of the phase Pi' 9 • • • 2n" 8.t the time <", 2 18 CONSERVATION OF C. . .fp'dq^' . . . dq: = r. . .^r'dp^'.. . dq:\ (45) where the limits in the two cases are fonned by corresponding phases. When the integrations cover infinitely small vari- ations of the momenta and coordinates, we may regard P' and P" as constant in the integrations and write F^^. ..fdp^f . . . dqjf = P' r. ..Cdpi'f . . . dqj'. Now the principle of the conservation of extension-in-phase, which has been proved (viz., in the second demonstration given above) independently of any reference to an ensemble of systems, requires that the values of the multiple integrals in this equation shall be equal. This gives Fff = Ff. With reference to an important class of cases this principle may be enunciated as follows. When the differential equations of motion are exactly known^ but the constants of the integral equations imperfectly deter- minedy the coefficient of probability of any phase at any time is equal to the coefficient of probability of the corresponding phase at any other time. By corresponding phases are meant those which are calculated for different times from the same values of the arbitrary constants of the integral equations. Since the sum of the probabilities of all possible cases is necessarily imity, it is evident that we must have j...jFdpi... dqn = 1, (46) phases where the integration extends over all phases. This is indeed only a different form of the equation aU iV= / . . . / I>dpi . . . dq^j phases which we may regard as defining N. PROBABILITY OF PHASE. 19 The values of the coefficient and index of probability of phase, like that of the density-in-phase, are independent of the system of coordinates which is employed to express the distri- bution in phase of a given ensemble. In dimensions, the coefficient of probability is the reciprocal of an extension-in-phase, that is, the reciprocal of the nth power of the product of time and energy. The index of prob- ability is therefore affected by an additive constant when we change our units of time and energy. If the unit of time is multiplied by c^ and the unit of energy is multiplied by ^, , all indices of probability relating to systems of n degrees of freedom will be increased by the addition of n log c, + n log 0,. (47) CHAPTER 11. APPLICATION OF THE PRINCIPLE OF CONSERVATION OP EXTENSION-IN-PHASE TO THE THEORY OF ERRORS. Let us now proceed to combine the principle which has been demonstrated in the preceding chapter and which in its differ- ent appUcations and regarded from different points of view has been variously designated as the conservation of density- in-phase, or of extension-in-phase, or of probability of phase, with those approximate relations which are generally used in the * theory of errors.' We suppose that the differential equations of the motion of a system are exactly known, but that the constants of the integral equations are only approximately determined. It is evident that the probability that the momenta and cotlrdinates at the time t^ fall between the limits 'p^ and 'p^ + rf^i', ?i' and 9.\ + ^2i'» ®^'> ^^^y ^® expressed by the formula 6^ dp^ . . . dqj, (48) where 77' (the index of probability for the phase in question) is a function of the coordinates and momenta and of the time. Let Qiy Ply etc. be the values of the coordinates and momenta which give the maximum value to 77', and let the general value of 77' be developed by Taylor's theorem according to ascending powers and products of the differences p^ — Pj', qi — Qiy etc., and let us suppose that we have a suiB&cient approximation without going beyond terms of the second degree in these differences. We may therefore set u' = - Ff, (49) where c is independent of the differences p^' — P^', q^ — Cl^ etc., and J" is a homogeneous quadratic function of these THEORY OF ERRORS. 21 differences. The terms of the first degree vanish in virtue of the maximum condition, which also requires that F' must have a positive value except when all the differences men- tioned vanish. If we set <7=e% (60) we may write for the probability that the phase lies within the limits considered Ce-^ dpi' . . . dqj. (51) C is evidently the maximum value of the coefficient of proba- biUty at the time considered. In regaid to the degree of approximation represented by these formulae, it is to be observed that we suppose, as is usual in the * theory of errors,' that the determination (ex- plicit or implicit) of the constants of motion is of such precision that the coefficient of probability e"^ or Ce"^' is practically zero except for very small values of the differences Pi' — Pj', q^' — Qi, etc. For very smaU. values of these differences the approximation is evidently in general sufficient, for larger values of these differences the value of (7e~^' will be sensibly zero, as it should be, and in this sense the formula wiU represent the facts. We shall suppose that the forces to which the system is subject are functions of the cob'rdinates either alone or with the tune. The principle of conservation of probabiUty of phase will therefore apply, which requires that at any other time (t") the maximum value of the coefficient of probability shall be the same as at the time ^', and that the phase (Pi", Qi\ etc.) which has this greatest probability-coefficient, shall be that which corresponds to the phase (Pj', Qi\ etc.), i. «., which is calculated from the same values of the constants of the integral equations of motion. We may therefore write for the probability that the phase at the time f' falls within the limits p^' and p^" + dp^', qi and q^" + dq^, etc., Ce-^" dp^^^ . . . dqj\ (52) 22 CONSERVATION OF EXTENSION-IN-PHASE where C represents the same value as in the preceding formula, viz., the constant value of the maximum coeiB&cient of probability, and F^' is a quadratic function of the differences Pi' - Pi^ 2i" - Qi^ etc., the phase (P/', (?/' etc.) being that which at the time ^" corresponds to the phase (P^', Q^^ etc.) at the time If. Now we have necessarily f. . .fc€r^'dp^...dqJz:zC. ..Cce-^"dp^"...dq:' = l, (53) when the integration is extended over all possible phases. It will be allowable to set ± cx> for the limits of all the coor- dinates and momenta, not because these values represent the actual limits of possible phases, but because the portions of the integrals lying outside of the limits of all possible phases will have sensibly the value zero. With ± oo for limits, the equation gives ^=-^ = 1, (64) where /' is the discriminant * of F\ and /" that of F". This discriminant is therefore constant in time, and like C an abso- lute invariant in respect to the system of coordinates which may be employed. In dimensions, like (7*, it is the reciprocal of the 2nth power of the product of energy and time. Let us see precisely how the functions P' and F" are related. The principle of the conservation of the probability-coefficient requires that any values of the coordinates and momenta at the time If shall give the function P' the same value as the corre- sponding coordinates and momenta at the time ^" give to P". Therefore P" may be derived from P' by substituting for Pii • • • 9'n' t^eir values in terms of p-l^ . . . q^. Now we have approximately * This term is used to denote tlie determinant having for elements on the principal diagonal the coefficients of the squares in the quadratic function F% and for its other elements the halves of the coefficients of the products in-P'. AND THEORY OF ERRORS. 23 ^»' - ■^^' = ^ ^' - -^^"^ • • • + ^ ^«'"" " ^-"^ S'.'-Q.' = |^(i'i"-A")... + |^(?."-«.'0, (65) and as in -F" terms of higher degree than the second are to be neglected, these equations may be considered accurate for the purpose of the transformation required. Since by equation (83) the eluninant of these equations has the value unity, tiie discriminant of J"' will be equal to that of F\ as has already appeared from the consideration of the principle of conservation of probability of phase, which is, in fact, essen- tially the same as that expressed by equation (83). At the time ^', the phases satisfying the equation Ff = *, (56) where h is any positive constant, have the probability-coeffi- cient C e~* . At the time ^', the corresponding phases satisfy the equation F^f = k, (57) and have the same probability-coefficient. So also the phases within the limits given by one or the other of these equations are corresponding phases, and have probability-coefficients greater than Ce""*, while phases without these limits have less probability-coefficients. The probability that the phase at the time if falls within the limits i^' = A; is the same as the probability that it falls within the limits J"' = A; at the time f, since either event necessitates the other. This probability may be evaluated as follows. We may omit the accents, as we need only consider a single time. Let us denote the ex- tension-in-phase within the limits i^ = A; by J7, and the prob- ability that the phase falls within these limits by R^ also the extension-in-phase within the limits i^ = 1 by Uy We have then by definition U=^ j . . . j dpi . . . dq^f (58) 24 CONSERVATION OF EXTENSION-IN-PHASE F=k JJ =s / . . . jCe"' dpi. .. dq^, (69) -p=i 1^1 =/.../ rfpi .• . dq^. (60) But since jP is a homogeneous quadratic function of the differences Pi — -Pi> i>« — -P«> • • • fi'ii — C«» we have identically F=k J. . .Jd(pi^FO . ..d(q^--Q^) kF=k =J'. . .Jl^d{p^ - Pi) . . . d{q^ - ^,) F=l = ^"J*. ''Jd(pi-Fi)... d(q^ « ^0- That is U=:Ic^Ui, (61) whence c^CTrz U^nk^^dk. (62) But if A; varies, equations (58) and (59) give F—k+dk dUzzz j. .. I dpi.. .dq^ (63) F=k dli=^ C.. . Cce-'dpi ...dq^ (64) Since the fetctor (7e~^ has the constant value Ce"^ in the last multiple integral, we have rf^= C e^ dU - C U^ne-^ le^^ dk, ((^) iJ = - dTilne-* ^1 + * + 2^ + . . . + r^^) + const. (66) We may determine the constant of integration by the condition that It vanishes with h. This gives AND THEORY OF ERRORS. 25 (67) i2== C7 &iln - OJTiijje-*^! + A: + ^ + . . . + r^^Y We may determine the value of the constant U^ by the con- dition that iJ = 1 for A = oo. This gives (7 [7^ [w = 1, and iJ = l-^(l + A + ^... + j^), (68) It is worthy of notice that the form of these equations de- pends only on the number of degrees of freedom of the system, being in other respects independent of its dynamical nature, except that the forces must be functions of the coordinates either alone or with the time. If we write for the value of h which substituted inequation (68) will give ^ = J, the phases determined by the equation F's.k^^ (70) will have the following properties. The probability that the phase falls within the limits formed by these phases is greater than the probability that it falls within any other limits enclosing an equal extension-in-phase. It is equal to the probability that the phase falls without the same limits. These properties are analogous to those which in the theory of errors in the determination of a single quantity belong to values expressed by A±a^ when A is the most probable value, and a the * probable error.' CHAPTER m. APPLICATION OF THE PRINCIPLE OF CONSERVATION OF EXTENSION-IN-PHASE TO THE INTEGRATION OF THE DIFFERENTIAL EQUATIONS OF MOTION.* We have seen that the principle of conservation of exten- sion-in-phase may be expressed as a difiFerential relation be- tween the coordinates and momenta and the arbitrary constants of the integral equations of motion. Now the integration of the differential equations of motion consists in the determina- tion of these constants as functions of the coordinates and momenta with the time, and the relation afforded by the prin- ciple of conservation of extension-in-phase may assist us in this determination. It will be convenient to have a notation which shaU. not dis- tinguish between the coordinates and momenta. If we write r-^ . . . r^ for the coordinates and momenta, and a ... A as be- fore for the arbitrary constants, the principle of which we wish to avail ourselves, and which is expressed by equation (37), may be written ^^^7'"'?:^ = ^nc. (a,... h). (71) Let us first consider the case in which the forces are deter- mined by the coordinates alone. Whether the forces are * conservative ' or not is immaterial. Since the differential equations of motion do not contain the time (f) in the finite form, if we eliminate dt from these equations, we obtain 2 w — 1 equations in r^ , . . . r2„ and their differentials, the integration of which will introduce 2 w — 1 arbitrary constants which we shall call 5 • • • A. If we can effect these integrations, the * See Boltzmann: '' ZuBammenhang zwischen den S&tzen iiber das Yer- halten mehratomiger GaBmolecule mit Jacobi's Princip des letzten Multi- pUcators. Sitzb. der Wiener Akad.,Bd. LXni, Abth. n., S. 679, (1871). THEORY OF INTEGRATION. 27 remaining constant (a) will then be introduced in the final integration, (viz., that of an equation containing c?^,) and will be added to or subtracted from t in the integral equation. Let us have it subtracted from t. It is evident then that Moreover, since J, ... A and t — a are independent functions of r^ , . . . raw, ^^ latter variables are functions of the former. The Jacobian in (71) is therefore function of J, ... A, and t — a, and since it does not vary with t it cannot vary with a. We have therefore in the case considered, viz., where the forces are functions of the coordinates alone, ^^ = func.(.,...A). (73) Now let us suppose that of the first 2 n — 1 integrations we have accomplished all but one, determining 2 w — 2 arbitrary constants (say t?, ... A) as functions of r^ , . . . rg^, leaving h as well as a to be determined. Our 2 w — 2 finite equations en- able us to regard aU the variables r^ , . . . rjjn, and all functions of these variables as functions of two of them, (say r^ and rg,) with the arbitrary constants t?, . . . A. To determine 6, we have the following equations for constant values of (?,... A. - dv^ - dvn __ rfro = -;- da + ^7- do. da do whence ^^^db = -pdr, + pdr^ (74) d(a,o) da da ^ ^ Now, by the ordinary formula for the change of variables, / . . . / ^^^ ^1 da db dr^... dr^n =/.../ dri .. . dr^^ = I ... I J . ' " ' ^" da . , • dh J J a{aj ... A) J J a{a) ... A) aift) . . . r^J 28 CONSERVATION OF EXTENSION-IN--PHASE where the limits of the multiple integrals are formed by the same phases. Hence djri^r^) _ d(ri , ... r^n) d(c, ...h) d(a, b) d{ay . . . h) d^r^, . . . r^^y ^ ^ With the aid of this equation, which is an identity, and (72), we may write equation (74) in the form t^;" •••??> f"--^\ dft = ;,dn-nrfr^ (76) The separation of the variables is now easy. The differen- tial equations of motion give r^ and r^ in terms of r^, . . . r2„. The integral equations already obtained give c^ . . . h and therefore the Jacobian rf(t?, . . . h)ld(r^^ . . . r2„), in terms of the same variables. But in virtue of these same integral equations, we may regard functions of r ^ , . • . r2„ as functions of r^ and r^ with the constants t?, . . . A. If therefore we write the equation in the form d(ru • « «*'2n) ^1, _ d{a,...h) ^^"" d{6,...h) ^^ d(e, ...h) ^""^^ Cir) d{Tzy . . . ra,) d{Tz^ . . . r^n) the coefficients of dr^ and dr^ may be regarded as known func- tions of r^ and rj with the constants t?, . . . A. The coefficient of db is by (73) a function of J, ... A. It is not indeed a known function of these quantities, but since (?,... A are regarded as constant in the equation, we know that the first member must represent the differential of some function of 6, ... A, for which we may write b\ We have thus . . '^^' ^ d(e, !. . h) ^^"^ "" d{c^,...h) ^''^' (78) c^(r8, . . . r2») d(Tzj . . . r^ which may be integrated by quadratures and gives V as func- tions of r^, r2, ...(?,«.. A, and thus as function of r^, . . . r^^. This mtegration gives us the last of the arbitrary constants which are functions of the coordinates and momenta without the time. The final integration, which introduces the remain- AND THEORY OF INTEGRATION, 29 ing constant (a), is also a quadrature, since the equation to be integrated may be expressed in the foim dt := F {r{) dri. Now, apart from any such considerations as have been ad- duced, if we limit ourselves to the changes which take place in time, we have identically fa drx — f 1 dr^ = 0, and r^ and r^ are given in terms of r^, . . . r^^ by the differential equations of motion. When we have obtained 2 w — 2 integral equations, we may regard r^ and r^ as known functions of r^ and r,. The only remaining difficulty is in integrating iius equation, K the case is so simple as to present no diflBculty, or if we have the skill or the good fortune to perceive that the multiplier 1 d(e,...h) ' (79) rf(r8, . . . r^n) or any other, will make the first member of the equation an exact differential, we have no need of the rather lengthy con- siderations which have been adduced. The utility of the principle of conservation of extension-in-phase is that it sup- plies a ' multiplier ' which renders the equation integrable, and which it might be difficult or impossible to find otherwise. It will be observed that the function represented by V is a particular case of that represented by b. The system of arbi- trary constants a^b\ c . . .h has certain properties notable for simplicity. If we write V for h in (77), and compare the result with (78), we get ,f^%' ■ ' '•'■\, = 1. (80) d(a, b',c, , . ,h) ^ ^ Therefore the multiple integral /• • P: da dbi dc.dh (81) 80 CONSERVATION OF EXTENSION-IN-'PHASE taken within limits formed by phases regarded as contempo- raneous represents the extension-in-phase within those limits. The case is somewhat different when the forces are not de- termined by the coordinates alone, but are functions of the coordinates with the time. All the arbitrary constants of the integral equations must then be regarded in the general case as functions of r^, . . . fgn, and t. We cannot use the princi- ple of conservation of extension-in-phase until we have made 2 w — 1 integrations. Let us suppose that the constants J, ... A have been determined by integration in terms of r^, . . . r^, and f, leaving a single constant (a) to be thus determined. Our 2 w — 1 finite equations enable us to regard all the variables ^i» • • • ^2« ^ functions of a single one, say r^ For constant values of 6, ... A, we have m=^^da + ridt. (82) Now I ... I -^ — da dr^ . . • dr^ ^iz I . . . I dr^ . • • rf^in J J d(a, ... A) d(ra, . . . rj^) where the limits of the integrals are formed by the same phases. We have therefore dvi d(rij . , • r2») d(bf ... A) da "" (f(a, ... A) d(r2, . . . r^^ * (83) by which equation (82) may be reduced to the form d(ri, . . . rg^) 1 ri d(a,...h) ^''■" c^(^...A) ^*'^"" d{h,...h) ^^' (^) Now we know by (71) that the coefficient of da is a func- tion of a, . . . A. Therefore, as i, ... A are regarded as constant in the equation, the first number represents the differential AND THEORY OF INTEGRATION. 81 of a function of a, ... A, which we may denote by of. We have then 1 ri' ^'= ct(b,...h) '^'•^- d(J,,...h) ^*' (85) which may be integrated by quadratures. In this case we may say that the principle of conservation of extension-in- phase has supplied the * multiplier ' 1 d(b, ...h) (86) d(r2f • . . ^am) for the integration of the equation dri ^ridt = 0. (87) The system of arbitrary constants a', 5, ... A has evidently the same properties which were noticed in regard to the system a, h\. . , A. CHAPTER IV. ON THE DISTRIBUTION IN PHASE CALLED CANONICAL, IN WHICH THE INDEX OP PROBABILITY IS A LINEAR FUNCTION OF THE ENERGY. Let us now give our attention to the statistical equilibrium of ensembles of conservation systems, especially to those cases and properties which promise to throw light on the phenom- ena of theimodynamics. The condition of statistical equilibrium may be expressed in the form* where P is the coefficient of probability, or the quotient of the density-in-phase by the whole number of systems. To satisfy this condition, it is necessary and sufficient that P should be a function of the ^'s and j's (the momenta and coordinates) which does not vary with the time in a moving system. In all cases which we are now considering, the energy, or any function of the energy, is such a function. P = f unc. (c) will therefore satisfy the equation, as indeed appears identi- cally if we write it in the form There are, however, other conditions to which P is subject, which are not so much conditions of statistical equilibrium, as conditions implicitly involved in the definition of the coeffi- * See equationg (20), (41), (42), aIbo the paragraph foUowing equation (20). The positions of any external bodies which can affect the systems are here supposed uniform for aU the systems and constant in time. CANONICAL DISTRIBUTION. 83 dent of probability, whether the case is one of equilibrium or not. These are: that P should be single- valued, and neither negative nor imaginary for any phase, and that ex- pressed by equation (46), viz., «n f" f PdPi ...dq^ — l. (89) phanon These considerations exclude P = € X constantf, as well as P = constant, as cases to be considered. The distribution represented by i, = logP = !^, (90) or where and yfr are constants, and positive, seems to repre- sent the most simple case conceivable, since it has the property that when the. system consists of parts with separate energies, the laws of the distribution in phase of the separate parts are of the same nature, — a property which enormously simplifies the discussion, and is the foundation of extremely important relations to thermodynamics. The case is not rendered less simple by the divisor 0, (a quantity of the same dimensions as €,) but the reverse, since it makes the distribution independent of the units employed. The negative sign of € is required by (89), which determines also the value of yfr for any given 0, viz., ^ all c = /.../ 6 ^dpi . . . dq^. (92) phaaes When an ensemble of systems is distributed in phase in the manner described, i. e., when the index of probability is a 8 e « 34 CANONICAL DISTRIBUTION linear function of the energy, we shall say that the ensemble is canonically distributed^ and shall call the divisor of the energy (0) the modulus of distribution. The fractional part of an ensemble canonically distributed which lies within any given limits of phase is therefore repre- sented by the multiple integral /• • •/ e 6 dpi . , . dq^ (93) taken within those lunits. We may express the same thing by saying tiiat the multiple mtegral expresses tiie probabiUty that an unspecified system of the ensemble (i. e., one of which we only know that it belongs to the ensemble) falls within the given limits. Since the value of a multiple integral of the form (23) (^which we have called an extension-in-phase) bounded by any given phases is independent of the system of coordinates by which it is evaluated, the same must be true of the multiple integral in (92), as appears at once if we divide up tiiis integral into parts so small that the exponential factor may be regarded as constant in each. The value of -^ is therefore in- dependent of the system of coordinates employed. It is evident that yfr might be defined as the energy for which the coefficient of probability of phase has the value unity. Since however this coefficient has the dimensions of the inverse nth power of the product of energy and time,* the energy represented by yjr is not independent of the units of energy and time. But when these units have been chosen, the definition of -^ will involve the same arbitrary constant as €, so that, while in any given case the numerical values of ^|r or € will be entirely indefinite until the zero of energy has also been fixed for the system considered, the difference ^Jr — e wiU represent a perfectly definite amount of energy, which is entirely independent of the zero of energy which we may choose to adopt. * See Chapter I, p. 19. OF AN ENSEMBLE OF SYSTEMS. 85 It ia^vident that the canonical distribution, is entirely deter- mined by the modulus (considered as a quantity of energy) and the nature of the system considered, since when equation (92) is satisfied the value of the multiple integral (93) is independent of the units and of the coordinates employed, and of the zero chosen for the energy of the system. In treating of the canonical distribution, we shall always suppose the multiple integral in equation (92) to have a finite value, as otherwise the coefficient of probability van- ishes, and the law of distribution becomes illusory. This wiU exclude certain cases, but not such apparently, as will affect the value of our results with respect to their bearing on ther- modynamics. It wiU exclude, for instance, cases in which the system or parts of it can be distributed in unlimited space (or in a space which has limits, but is still infinite in volume), while the energy remains beneath a finite limit. It also excludes many cases in which the energy can decrease without limit, as when the system contains material points which attract one another inversely as the squares of their distances. Cases of material points attracting each other inversely as the distances would be excluded for some values of 0, and not for others. The investigation of such points is best left to the particular cases. For the purposes of a general discussion, it is sufficient to caU attention to the assumption implicitly involved in the formula (92).* The modulus has properties analogous to those of tem- perature in thermodynamics. Let the system A be defined as one of an ensemble of systems of m degrees of freedom distributed in phase with a probability-coefficient e e , * It wiU be observed that similar limitations exist in thermodynamics. In order that a mass of gas can be in thermodynamic equilibrium, it is necessary that it be enclosed. There is no thermodynamic equilibrium of a (finite) mass of gas in an infinite space. Again, that two attracting particles should be able to do an infinite amount of work in passing from one configuration (which is regarded as possible) to another, is a notion which, although per- fectly intelligible in a mathematical formula, is quite foreign to our ordinary conceptions of matter. 36 CANONICAL DISTRIBUTION and the system B as one of an ensemble of systems of n degrees of freedom distributed in phase with a probability- coefficient e e , which has the same modulus. Let q^^ . . .q^^ i>i> • • • i>m be the coordinates and momenta of A^ and q,^^ , . . . y^^^, jp^^.^ , . . . p,j^^ those of B. Now we may regard the systems A and B as together forming a system C, having m + n degrees of free- dom, and the coordinates and momenta g'l, . . . ?,»+«, jPi» • • • Pm^* The probability that the phase of the system C, as thus defined, will fall within the limits dpi, ... dj^m^y dqi, .. . dqj„^ is evidently the product of the probabilities that the systems A and B will each fall within the specified Kmits, viz., e ® dpi. . . dp„^ dqi. . . dq„^. We may therefore regard C as an undetermined system of an ensemble distributed with the probabiHtyKJoefficient e (96) an ensemble which might be defined as formed by combining each system of the first ensemble with each of the second. But since €^ + e^ is the energy of the whole system, and 1^^ and yfr^ are constants, the probability-coefficient is of the general form which we are considering, and the ensemble to which it relates is in statistical equilibrium and is canonically distributed. This result, however, so far as statistical equilibrium is concerned, is rather nugatory, since conceiving of separate systems as forming a single system does not create any in- teraction between them, and if the systems combined belong to ensembles in statistical equilibrium, to say that the ensemble formed by such combinations as we have supposed is in statis- tical equilibrium, is only to repeat the data in different OF AN ENSEMBLE OF SYSTEMS. 37 words. Let us therefore suppose that in forming the system G we add certain forces acting between A and B, and having the force-function — €j^. The energy of the system is now €a + ^3 + ^AB9 3nd an ensemble of such systems distributed with a denBity proportional to would be in statistical equilibrium. Comparing this with the probability-coeflScient of given above (95), we see that if we suppose €j^ (or rather the variable part of this term when we consider all possible configurations of the systems A and B) to be infinitely small, the actual distribution in phase of C win differ infinitely little from one of statistical equilibriimi, which is equivalent to saying that its distribution m phase will vary infinitely little even in a time indefinitely prolonged.* The case would be entirely different if A and B belonged to ensembles having different moduli, say 0^ and 0^. The prob- ability-coeflScient of C would then be ,-e- + ^r, (97) which is not approximately proportional to any expression of the form (96). Before proceeding farther in the investigation of the dis- tribution in phase which we have called canonical, it will be interesting to see whether the properties with respect to * It wiU be observed that the above condition relating to the forces which act between the different systems is entirely analogous to that which must hold in the corresponding case in thermodynamics. The most simple test of the equality of temperature of two bodies is that they remain in equilib- rium when brought into thermal contact. Direct thermal contact implies molecular forces acting between the bodies. Now the test will fail unless the energy of these forces can be neglected in comparison with the other energies of the bodies. Thus, in the case of energetic chemical action be- tween the bodies, or when the number of particles affected by the forces acting between the bodies is not negligible in comparison with the whole number of particles (as when the bodies have the form of exceedingly thin sheets), the contact of bodies of the same temperature may produce con- siderable thermal disturbance, and thus fail to afford a reliable criterion of the equality of temperature. 38 OTHER DISTRIBUTIONS statistical equilibrium which have been described are peculiar to it, or whether other distributions may have analogous properties. Let 71^ and ri" be the indices of probability in two independ- ent ensembles which are each in statistical equilibrium, then 17' + ly" will be the index in the ensemble obtained by combin- ing each system of the first ensemble with each system of the second. This third ensemble will of course be in statistical equilibrium, and the function of phase 17' + rf^ will be a con- stant of motion. Now when infinitesimal forces are added to the compound systems, if 17' + 17" or a function differing infinitesimally from this is still a constant of motion, it must be on account of the nature of the forces added, or if their action is not entirely specified, on account of conditions to which they are subject. Thus, in the case already considered, 7}' + 97" is a function of the energy of the compound system, and the infinitesimal forces added are subject to the law of conservation of energy. Another natural supposition in regard to the added forces is that they should be such as not to affect the moments of momentum of the compound system. To get a case in which moments of momentum of the compound system shall be constants of motion, we may imagine material particles con- tained in two concentric spherical shells, being prevented from passing the surfaces bounding the shells by repulsions acting always in lines passing through the common centre of the shells. Then, if there are no forces acting between particles in different shells, the mass of particles in each shell wiU have, besides its energy, the moments of momentum about three axes through the centre as constants of motion. Now let us imagine an ensemble formed by distributing in phase the system of particles in one shell according to the index of probability where € denotes the energy of the system, and o)^, 0)29 ^39 its three moments of momentum, and the other letters constants. HAVE ANALOGOUS PROPERTIES. 39 In like manner let us imagine a second ensemble formed by distributing in phase the system of particles in the other shell according to the index A'^i + '!^ + !!!L + ^^ (99) where the letters have similar significations, and 0, il^, ilg* ^s the same values as in the preceding formula. Each of the two ensembles will evidently be in statistical equilibrium, and therefore also the ensemble of compound systems obtained by combining each system of the first ensemble with each of the second. In this third ensemble the index of probability will be A..A'-'-±i + ^ + r:i±^+^, (100) where the four numerators represent functions of phase which are constants of motion for the compound systems. Now if we add in each system of this third ensemble infiiii- tesimal conservative forces of attraction or repulsion between particles in different shells, determined by the same law for all the systems, the functions ©^ + ©', ©g + a>2i ^^^ ®8 + W wiU remain constants of motion, and a function differing in- finitely little from 6^ + e' will be a constant of motion. It would therefore require only an infinitesimal change in the distribution in phase of the ensemble of compound systems to make it a case of statistical equilibrium. These properties are entirely analogous to those of canonical ensembles.* Again, if the relations between the forces and the coordinates can be expressed by linear equations, there wiU be certain "normal" types of vibmtion of which the actual motion may be regarded as composed, and the whole energy may be divided * It would not be possible to omit the term relating to energy in the above indices, since without this term the condition expressed bj equation (89) cannot be satisfied. The consideration of the above case of statistical equilibrium may be made the foundation of the theory of the thermodynamic equilibrium of rotating bodies, — a subject which has been treated by Maxwell in his memoir *' On Boltzmann's theorem on the average distribution of energy in a system of material points/' Cambr. FhU. Trans., voL XTT, p. 647, (1878). 40 OTHER DISTRIBUTIONS into parts relating separately to vibrations of these different types. These partial energies wUl be constants of motion, a^d if such a system is Ltributed according to an index which is any function of tixe partial energies, the ensemble wiU be in statistical equilibrium. Let the index be a linear func- tion of the partial energies, say Let us suppose that we have also a second ensemble com- posed of systems in which the forces are linear functions of the cotJrdinates, and distributed in phase according to an index which is a linear function of the partial energies relating to the normal types of vibration, say A'--^,...-^^. (102) Since the two ensembles are both in statistical equilibrium, the ensemble formed by combining each system of the first with each system of the second will also be in statistical equilibrium. Its distribution in phase wiU be represented by the index ^ + ^' - ^ • • • - ^ - ^ • • • -" ^ > (103) and the partial energies represented by the numerators in the formula will be constants of motion of the compound systems which form this third ensemble. Now if we add to these compound systems infinitesimal forces acting between the component systems and subject to the same general law as those already existing, viz., that they are conservative and linear functions of the coordinates, there will still be w + w types of normal vibration, and n + m partial energies which are independent constants of motion. If all the original n -i- m normal types of vibration have differ- ent periods, the new types of normal vibration will differ infini- tesimally from the old, and the new partial energies, which are constants of motion, will be nearly the same functions of phase as the old. Therefore the distribution in phase of the HAVE ANALOGOUS PROPERTIES. 41 ensemble of compound systems after the addition of the sup- posed infinitesimal forces wiU differ infinitesimally from one which would be in statistical equilibrium. The case is not so simple when some of the normal types of motion have the same periods. In this case the addition of infinitesimal forces may completely change the normal types of motion. But the sum of the partial energies for all the original types of vibration which have any same period, wiU be nearly identical (as a function of phase, i. e., of the coordi- nates and momenta,) with the sum of the partial energies for the normal types of vibration which have the same, or nearly the same, period after the addition of the new forces. If, therefore, the partial energies in the indices of the first two ensembles (101) and (102) which relate to types of vibration having the same periods, have the same divisors, the same will be true of the index (103) of the ensemble of compound sys- terns, and the distribution represented wiU differ infinitesimally from one 'which would be in statistical equilibrium after the addition of the new forces.* The same would be true if in the indices of each of the original ensembles we should substitute for the term or terms relating to any period which does not occur in the other en- semble, any function of the total energy related to that period, subject only to the general limitation expressed by equation (89). But in order that the ensemble of compound systems (with the added forces) shall always be approximately in statistical equilibrium, it is necessary that the indices of the original ensembles should be linear functions of those partial energies which relate to vibrations of periods common to the two ensembles, and that the coefficients of such partial ener- gies should be the same in the two mdices.f * It is interesting to compare the above relations with the laws respecting the exchange of energy between bodies bj radiation, although the phenomena of radiations lie entirely without the scope of the present treatise, in which the discussion is limited to systems of a finite number of degrees of freedom. t The above may perhaps be sufficiently illustrated by the simple case where n = 1 in each system. If the periods are different in the two systems, they may be distributed according to any functions of the energies : but if 42 CANONICAL DISTRIBUTION The properties of canonically distributed ensembles of systems with respect to the equilibrium of the new ensembles which may be formed by combining each system of one en- semble with each system of another, are therefore not peculiar to them in the sense that analogous properties do not belong to some other distributions under special Umitations in regard to the systems and forces considered. Yet the canonical distribution evidently constitutes the most simple case of the kind, and that for which the relations described hold with the least restrictions. Returning to the case of the canonical distribution, we shall find other analogies with thermodynamic systems, if we suppose, as in the preceding chapters,* that the potential energy (e^) depends not only upon the coordinates ^i • • • ^n which determine the configuration of the system, but also upon certain coordinates ai, as, etc. of bodies which we call external^ meaning by this simply that they are not to be re- garded as forming any part of the system, although their positions affect the forces which act on the system. The forces exerted by the system upon these external bodies will be represented by — deqjda^, — dcqlda^^ etc., while — dcg/dq^j . . . — dcg/dq^ represent all the forces acting upon the bodies of the system, including those which depend upon the position of the external bodies, as well as those which depend only upon the configuration of the system itself. It will be under- stood that €p depends only upon 91 , . . . y„ , |?i , . . . j9„ , in other words, that the kinetic energy of the bodies which we call external forms no part of the kinetic energy of the system. It follows that we may write * =|^=-^i, (104) dai doi although a similar equation would not hold for differentiations relative to the internal coordinates. the periods are the same they must be distributed canonicaUy with same modulus in order that the compoimd ensemble with additional forces may be in statistical equilibrium. * See especially Chapter I, p. 4. e ® OF AN ENSEMBLE OF SYSTEMS. 43 We always suppose these external coordinates to have the same values for aU systems of any ensemble. In the case of a canonical distribution, f. «., when the index of probability of phase is a linear function of the energy, it is evident that the values of the external coordinates will affect the distribu- tion, since they affect the energy. In the equation = /.../e ^dpi...dq^, (105) phaaes by which yfr may be determined, the external coordinates, a^, ^2, etc., contained implicitly in e, as weU as 0, are to be re- garded as constant in the integrations indicated. The equa- tion indicates that i^ is a function of these constants. If we imagine their values varied, and the ensemble distributed canonically according to their new values, we have by differentiation of the equation ^ / \ all c phases aU - __^ phases -• aU , * 1 C P dc — ~0^^M-"/^^ ® c^Pi . . . c^S'n - etc., (106) phases or, multiplying by e®, and setting * - y* - — - X etc aU ^^— g -d^ + ^dJ0 = -c^© / .. . I €6 ® dpi...dq^ phases •11 ^ ♦'^ + dai C . . jAi e ® dpi . . . dq. phases •U ^-€ + da^j ' • • I A^e ^ dpi. . .dq^ + etc. (107) phases 44 CANONICAL DISTRIBUTION Now the average value in the ensemble of any quantity (which we shall denote in general by a horizontal line above the proper symbol) is determined by the equation /all ^ ^-€ .'.Jue^ dpi... dq^. (108) phftMs Comparing this with the preceding equation, we have dip = ^d® — ^d® — Ji doi — Jj da, — etc. (109) Or, since 5^ = v, (HO) and ^ = V, (111) d\l/ = ^d® — Ai dai — iTj da^ — etc. (112) Moreover, since (111) gives dif/ ^d€=:®d^ + ^d®, (113) we have also cKi = — c^^ — JTi doi — -ii da2 — etc. (H^) This equation, if we neglect the sign of averages, is identi- cal in form with the thermodynamic equation , de + Ai dai + A2 da^ + etc. ,-t^i^ dv = ji > (llo) or d€= Tdrj — Ai dai — A2 da^ — etc., (H^) which expresses the relation between the energy, tempera- ture, and entropy of a body in thermodynamic equilibrium, and the forces which it exerts on external bodies, — a relation which is the mathematical expression of the second law of thermodynamics for reversible changes. The modulus in the statistical equation corresponds to temperature in the thermo- dynamic equation, and the average index of probability with its sign reversed corresponds to entropy. But in the thermo- dynamic equation the entropy (ly) is a quantity which is OF AN ENSEMBLE OF SYSTEMS. 45 only defined by the equation itseK, and incompletely defined in that the equation only determines its differential, and the constant of integration is arbitrary. On the other hand, the ^ in tte Bto«S «„.ao» CLa c^npMy de«ned a, the average value in a canonical ensemble of systems of the logariLn of the coefficient of probability of phase. We may also compare equation (112) with the thermody- namic equation 1^ = — rjdT— Aidtti — A^doz — etc., (117) where yjr represents the function obtained by subtracting the product of the temperature and entropy from the energy. How far, or in what sense, the similarity of these equations constitutes any demonstration of the thermodynamic equa- tions, or accounts for the behavior of material systems, as described in the theorems of thermodynamics, is a question of which we shall postpone the consideration until we have furtiier investigated tiie properties of an ensemble of systems distributed in phase accord^g to tiie law which we ai con- sidering. The analogies which have been pointed out will at least supply the motive for this investigation, which will naturally commence with the determination of the average values in the ensemble of the most important quantities relating to the systems, and to the distribution of the ensemble with respect to the different values of these quantities. CHAPTER V. AVERAGE VALUES IN A CANONICAL ENSEMBLE OF SYSTEMS. In the simple but important case of a system of material points, if we use rectangular coordinates, we have for the product of the differentials of the coordinates dxi dj/i dzi . . . dxy dt/y dz^, and for the product of the differentials of the momenta • • • • • • fTii doci THi dt/i rrii dzi . . . THy dXy niy dyy nty dzy • The product of these expressions, which represents an element of extension-in-phase, may be briefly written m\ dxi . . • niy dzy dxi • • • dzy ; and the integral I • . . I 6 ® Wi dxi . » . my dzy dxi . . . dzy (118) will represent the probability that a system taken at random from an ensemble canonically distributed will fall within any given limits of phase. In this case € = e^ + J mi cci* . . . + ^myZy\ (H^) and ^e 4f-eg miXi^ m^^ e ® =e ® e 28 . ..e 28 ^ (120) The potential energy (e^) is independent of the velocities, and if the limits of integration for the co^irdinates are inde- pendent of the velocities, and the limits of the several veloci- ties are independent of each other as well as of the cotJrdinates, VALUES IN A CANONICAL ENSEMBLE. 47 the multiple integral may be resolved into the product of integrals f'"fe ® dxi. . .dzy j 6 ^® midxi . . . j e ^® niydzy. (121) This shows that the probability that the configuration lies within any given limits is independent of the velocities, and that tiie probability that any component velocity lies within any giVfen limits is independent of the other component velocities and of the configuration. Since and I 6 ^® Wi dxi = V2 «• ^1 ®9 t/ — 00 (122) +« -"i^i' r — 00 irrnxi^e ^® Wi efoji = V J 'r ^i ®», 0^^) the average value of the part of the kinetic energy due to the velocity x^, which is expressed by the quotient of these inte- grals, is ^ 0. This is true whether the average is taken for the whole ensemble or for any particular configuration, whether it is taken without reference to the other component velocities, or only those systems are considered in which the other component velocities have particular values or lie within specified limits. The number of codrdinates is 3 1/ or n. We have, therefore, for the average value of the kinetic energy of a system €p = fv© = Jn0. (124) This is equally true whether we take the average for the whole ensemble, or limit the average to a single configuration. The distribution of the systems with respect to their com- ponent velocities follows the ' law of errors ' ; the probability that the value of any component velocity lies within any given limits being represented by the value of the corresponding integral in (121) for those limits, divided by (2 7rwi0)*, 48 AVERAGE VALUES IN A CANONICAL which is the value of the same integral for infinite limits. Thus the probability that the value of x^ lies between any given limits is expressed by (2^0) J ^ r/iix^ 8 6 ^® (&i. (126) The expression becomes more simple when the velocity is expressed with reference to the energy involved. If we set the probability that 8 lies between any given limits is expressed by Here 8 is the ratio of the component velocity to that which would give the energy ; in other words, «2 is the quotient of the energy due to the component velocity divided by 0. The distribution with respect to the partial energies due to the component velocities is therefore the same for all the com- ponent velocities. The probability that the configuration lies within any given limits is expressed by the value of M* (27r®)2 r... Ce ^ dx^. ..dz^ (127) for those limits, where M denotes the product of aU the masses. This is derived from (121) by substitution of the values of the integrals relating to velocities taken for infinite limits. Very similar results may be obtained in the general case of a conservative system of n degrees of freedom. Since e^ is a homogeneous quadratic function of the jp's, it may be divided into parts by the formula e, = i2>.g... + ii'.^ (128) ENSEMBLE OF SYSTEMS. 49 where € might be written for e^ in the differential coefficients without affecting the signification. The average value of the first of these parts, for any given configuration, is expressed by the quotient ft:, r %) —00 t/ — ( J^i -^ — e dpi . • . dpfi n:r- de 00 J' dpi 00 fci (129) 6 ® dp^... dp. ^ e 00 Now we have by integration by parts By substitution of this value, the above quotient reduces to 75-, which is therefore the average value of hPi-y^ for the Z dpx given configuration. Since this value is independent of the configuration, it must also be the average for the whole ensemble, as might easily be proved directly. (To make the preceding proof apply directly to the whole ensemble, we have only to write dp^ . . . dg',, for dp^ . . . dp, in the multiple integrals.) This gives J ?i for the average value of the whole kinetic energy for any given configuration, or for the whole ensemble, as has already been proved in the case of material pomts. The mechanical significance of the several parts into which the kinetic energy is divided in equation (128) will be appar- ent if we imagine that by the application of suitable forces (different from those derived from e^ and so much greater that the latter may be neglected in comparison) the system was brought from rest to the state of motion considered, so rapidly that the configuration was not sensibly altered during the process, and in such a manner also that the ratios of the component velocities were constant in the process. If we write Fid(ix . . . + F,dq, 4 60 AVERAGE VALUES IN A CANONICAL for the moment of these forces, we have for the period of their action by equation (8) The work done by the force F^ may be evaluated as follows : where the last term may be cancelled because the configuration does not vary sensibly during the application of the forces. (It will be observed that the other terms contain factors which increase as the time of the action of the forces is diminished.) We have therefore, Jf^ dq^ =Jp^ qi dt =Jq^ dp =^^fpi <^Pi • (l^l) For since the ^'s are linear functions of the j's (with coeffi- cients involving tlie j's) the supposed constancy of the j's and of the ratios of the g''s will make the ratio Qi/pi constant. The last integral is evidently to be taken between the limits zero and the value of p^ in the phase originally considered, and the quantities before the integral sign may be taken as relating to that phase. We have therefore Jf, dq^ = ipi y\ = J^i ^ . (132) That is: the several parts into which the kinetic energy is divided in equation (128) represent the amounts of energy communicated to the system by the several forces J\, . . . -F„ under the conditions mentioned. The following transformation will not only give the value of the average kinetic energy, but will also serve to separate the distribution of the ensemble in configuration from its dis- tribution in velocity. Since 2 e^ is a homogeneous quadratic function of the j^'s, which is incaj^ble of a negative value, it can always be ex- pressed (and in more than one way) as a sum of squares of ENSEMBLE OF SYSTEMS. 61 linear functions of the p^&.* The coefficients in these Knear functions, like those in the quadratic function, must be regarded in the general case as functions of the j's. Let 2 €p = wi« + wa^ . . . + V (133) where u^ ... u^^ axe such linear functions of the jt?'s. K we write d(pi . . . i?n) for ihe Jacobian or determinant of the differential coefficients of the form dp/du, we may substitute d(Ui . . . i/„) for dp^ . . . dpn under the multiple integral sign in any of our formulae. It will be observed that this determinant is function of the j's alone. The sign of such a determinant depends on the rela- tive order of the variables in the numerator and denominator. But since the suffixes of the w's are only used to distinguish these functions from one another, and no especial relation is supposed between a p and a u which have the same suffix, we may evidently, without loss of generality, suppose the suffixes so applied that the determinant is positive. Since the w's are linear functions of the j^'s, when the in- tegrations are to cover all values of the j^'s (for constant j's) once and only once, they must cover all values of the ti's once and only once, and the limits will be ± oo for all the w's. Without the supposition of the last paragraph the upper limits would not always be + oo , as is evident on considering the effect of changing the sign of a u. But with the supposition which we have made (that the determinant is always positive) we may make the upper limits + oo and the lower — oo for all the w's. Analogous considerations will apply where the in- tegrations do not cover all values of the p's and therefore of * The reduction requires only the repeated application of the process of 'completing the square' used in the solution of quadratic equations. 52 AVERAGE VALUES IN A CANONICAL the t^'s. The integrals may always be taken from a less to a greater value of a u. The general integral which expresses the fractional part of the ensemble which falls within any given lunits of phase is thus reduced to the form /•••/• For the average value of the part of the kinetic energy which is represented by ^ u^\ whether the average is taken for the whole ensemble, or for a given configuration, we have therefore ""-= r+^.«^ — =l2^=r ^'''> and for the average of the whole kinetic energy, \n%^ as before. The fractional part of the ensemble which lies within any given limits of configuration^ is found by integrating (134) with respect to the t^'s from — oo to + oo . This gives «L — f which shows that the value of the Jacobian is independent of the manner in which 26^ is divided into a sum of squares. We may verify this directly, and at the same time obtain a more convenient expression for the Jacobian, as follows. It will be observed that since the i^'s are linear functions of the j^'s, and the j^'s linear functions of the y's, the w's will be linear functions of the j's, so that a differential coefficient of the form duldq wiU be independent of the ?'s, and function of the g''s alone. Let us write dpjduy for the general element of the Jacobian determinant. We have ENSEMBLE OF SYSTEMS. 68 dpgg d d€ d '^ d€ dUf. dUy dUg dq^ du^ r=i du^ dq^ — ^ ( ^^ du\ __ d^ d€^ ^ du^ r=i \du^ du^ dqj dq^, du^ dq^ (^^') Therefore d(jp, . . .pn) _ djuy ■ * . u,) d{uy ... 1*0 d(2y *' •in) ^^^ and These determinants are all functions of the j's alone.* The last is evidently the Hessian or determinant formed of the second differential coefficients of the kinetic energy with re- spect to ji , . . . y„. We shall denote it by A j. The reciprocal determmant d(qi » > « g>) d(pi . . . Pnf which is the Hessian of the kinetic energy regarded as func- tion of the ^*s, we shall denote by A,. K we set e ^ =z I . . . I tf ® A, dpy ... dp^ —00 -H» _+« — ^1^ » ■ » — «m^ ft e 28 ciwi . . . (fw^ = (2^0)2 (140) =/••/ and ^, = ,^ — ,^p, (141) * It will be obserred that the proof of (137) dependfl on the linear relation between the ti's and q% which makes — r- constant with respect to the differ- aqx entiations here considered. Compare note on p. 12. 64 AVERAGE VALUES IN A CANONICAL the fractional part of the ensemble which lies within any given limits of configuration (186) may be written I ... I e e ^- dqi. . . dq^, (142) where the constant ^jr^ may be determined by the condition that the integral extended over all configurations has the value imity.* * In the simple but important case in which A^ is independent of the q% and €q a quadratic function of the q% if we write ca for the least value of €, (or of ff) consistent with the given values of the external coordinates, the equation determining ^g may be written e = A * / . • . / « dqi . . . dqn, — 00 —00 If we denote by qi, ...qn the values of 9i, . . . 9n which give c^ its least value ffo) it is evident that Cf — ca is a homogenous quadratic function of the differ- ences qi — qi, etc., and that dqi^ . . . dqn may be regarded as the differentials of these differences. The evaluation of this integral is therefore analytically similar to that of the integral +00 +00 _'P J , . .J e dpi . . . dpnf for which we have found the value Ap'~*'(2tG)5. By the same method, or by analogy, we get where A^ is the Hessian of the potential energy as function of the ^s. It will be observed that Ag depends on the forces of the system and is independ- ent of the masses, while A^ or its reciprocal Ap depends on the masses and is independent of the forces. While each Hessian depends on the system of coordinates employed, the ratio A^/A^ is the same for all systems. Multiplying the last equation by (140), we have For the average value of the potential energy, we have +00 +00 *y~*g J " 'J (^« ■" *«) * ^^i • • • ^^* 5[ — €a = +00 +« *9~^a J • . . / 6 dq^^ . . . dq<^ ENSEMBLE OF SYSTEMS. 55 When an ensemble of systems is distributed in confyura- tion in the manner indicated in this formula, i. «., when its distribution in configuration is the same as that of an en- semble canonically distributed in phase, we shall say, without any reference to its velocities, that it is canonically distribtUed in configuration. For any given configuration, the fractional part of the systems which Ue within any given limits of velocity is represented by the quotient of the multiple integral r. . . jc e d[pi . . . dp^j or its equivalent /•••/ -u^^ . . . -ti«a e ^® Aj*(?««i ...du, nf taken within those limits divided by the value of the saine integral for the limits ± oo. But the value of the second multiple integral for the limits ± oo is evidently A/(27r0)l "We may therefore write ' "J e ® dtii... dUn, (143) The eyaluation of this expression is similar to thd,t of +« +00 _i -f-OO +00 P, y, . .ye dpi . . . dpn — 00 which expresses the average yalue of the kinetic energy, and for which we haye f onnd the yalue 5 n 6. We haye accordingly «fl — «a = 2'*®' Adding the equation we haye 7 — t a = n e. 66 AVERAGES IN A CANONICAL ENSEMBLE. "J 6 » i^^dp^...dp,, (144) or again T' ' ' A ^ ^i^^i • • • ^.^ (145) for the fractional part of the systems of any given configura- tion which lie within given limits of velocity. When systems are distributed in velocity according to these formulae, i. «., when the distribution in velocity is like that in an ensemble which is canonically distributed in phase, we shall say that they are canonically distributed in velocity. The fractional part of the whole ensemble which falls within any given limits of phase, which we have before expressed in the form e ® dpx . . . dp^dqi . . . dq^^ (1^:6) /•••/■ may also be expressed in the form e ® ^idqi . . . dq^dqi . . . dq^. (147) / •/' CHAPTER VI. EXTENSION IN CONFIGURATION AND EXTENSION IN VELOCITY. The formulae relating to canonical ensembles in the closing paragraphs of the last chapter suggest certain general notions and principles, which we shall consider in this chapter, and which are not at all limited in their application to the canon- ical law of distribution.* We have seen in Chapter IV. that the nature of the distribu- tion which we have called canonical is independent of the system of coordinates by which it is described, being deter- mined entii'ely by the modulus. It follows that the value represented by the multiple integral (142), which is the frac- tional part of the ensemble which lies within certain limiting configurations, is independent of the system of coordinates, being determmed entirely by the Imiitmg configurations with the modulus. Now •^, as we have already seen, represents a value which is independent of the system of coordinates by which it is defined. The same is evidently true of i^p ^7 equation (140), and therefore, by (141), of yjrg. Hence the exponential factor in the multiple integral (142) represents a value which is independent of the system of coordinates. It follows that the value of a multiple integral of the form r. . . Ct^^dq^ . . . ^« (148) * These notions and principles are in fact such as a more logical arrange- ment of the subject would place in connection with those of Chapter I., to which they are closely related. The strict requirements of logical order have been sacrificed to the natural development of the subject, and very elementary notions have been left untU they have presented themselves in the study of the leading problems. 68 EXTENSION IN CONFIGURATION is independent of the system of coordinates which is employed for its evaluation, as will appear at once, if we suppose the multiple integral to be broken up into parts so small that the exponential factor may be regarded as constant in each. In the same way the formulae (144) and (145) which express the probability that a system (in a canonical ensemble) of given configuration will fall within certain limits of velocity, show that multiple integrals of the form J...Jt^dp^...dp, (149) (150) or J "J Ai dqi ...d^n relating to velocities possible for a given configuration, when the limits are formed by given velocities, have values inde- pendent of the system of coordinates employed. These relations may easily be verified directly. It has al- ready been proved that d(Pi, ...Pn) _ d(qi . . .q'n) _ d(qi, . . . g,) d(pi,...pn) d(Qi,...Q^) d(Qi,,..Q^) where Ji » • • • y«ii?i , . . .p^ and ^^ , . . . ^„, Pj , . . . P„ are two systems of coordinates and momenta.* It follows that J J \d{qi, . . . 9'») / J J \d(qu . . . qn)J d(Qi, ...Qn) * See equation (29). AND EXTENSION IN VELOCITY. 69 and J J \d(P^,...F^)J \d(pr,...pn)J \d(Qi,...Qn)J ^' J J \d(jPl,'"Pn)J 1 • • • dp^' The multiple integral f . . . / dpi • • . dp^dqi • . • c^s^M) (1^1) which may also be written I ... I AjdJg'i . . . dq^dqi , . . dq^^, (152) and which, when taken within any given limits of phase, has been shown to have a value independent of the coordinates employed, expresses what we have called an extenaion-in- phase.* In like manner we may say that the multiple integral (148) expresses an extension-in-configuratioTiy and that the multiple integrals (149) and (150) express an extensiortrinr velocity. We have called dpi . . . dp^dqi , . , dq^, (163) which is equivalent to ^qdqi . • . dq^dqi . . . dq^, (164) an element of extension-in-phase. We may caU ^'Mqi ...dq^ (165) an element of extension-in-configuration, and Ap*(^i?i . . . dp^j (166) * See Chapter 1, p. 10. 60 EXTENSION IN CONFIGURATION or its equivalent Ai*c£gr'i ...64^, (167) an element of extension-in-velocity. An extension-in-phase may always be regarded as an integral of elementary extensions-in-configuration multiplied each by an extension-in-velocily. This is evident from the formulae (151) and (162) which express an extension-in-phase, if we imagine the integrations relative to velocity to be first carried out. The product of the two expressions for an element of extension-in-velocity (149) and (150) is evidently of the same dimensions as the product that is, as the nth power of energy, since every product of the form p^ qi has the dimensions of energy. Therefore an exten- sion-in-velocity has the dimensions of the square root of the nth power of energy. Again we see by (155) and (156) that the product of an extension-in-configuration and an extension- in-velocity have the dimensions of the nth. power of energy multiplied by the wth power of time. Therefore an extension- in-configuration has the dimensions of the nth power of time multipUed by the square root of the nth power of energy. To the notion of extension-in-configuration there attach themselves certain other notions analogous to those which have presented themselves in connection with the notion of ex- tension-in-phase. The number of systems of any ensemble (whether distributed canonically or in any other manner) which are contained in an element of extension-in-configura- tion, divided by the numerical value of that element, may be called the dermty-irirconfiguration. That is, if a certain con- figuration is specified by the coordinates jj • . . ?„, and the number of systems of which the coordinates fall between the limit3 Ji and Ji + ci Ji , . . . ?„ and y„ + dq^ is expressed by A^i^^S'i • • • dq,^ (158) AND EXTENSION IN VELOCITY. 61 2), wiU be the den8it7-in-con%uration. AM if we set e" = ^, (159) where iV denotes, as usual, the total number of systems in the ensemble, the probability that an unspecified system of the ensemble will Ml within the given limits of configuration, is expressed by e'^^^'^dqi ...dq^. (160) We may call e'* the coefficient of prohalility of the configurch tiorij and rjg the index of probability of the configuration. The fractional part of the whole number of systems which are within any given limits of configuration will be expressed by the multiple integral j. . . ie'^^Ai^dqi . . . dq^. (161) The value of this integral (taken within any given configura- tions) is therefore independent of the system of coordinates which is used. Since the same has been proved of the same integral without the factor e\ it follows that the values of rjg and Dq for a given configuration in a given ensemble are independent of the system of coordinates which is used. The notion of extension-in-velocity relates to systems hav- ing the same configuration.* If an ensemble is distributed both in configuration and in velocity, we may confine our attention to those systems which are contained within certain infinitesimal limits of configuration, and compare the whole number of such systems with those which are also contained * Except Id some simple cases, such as a system of material points, we cannot compare velocities in one configuration with yelocities in another, and speak of their identity or difference except in a sense entirely artificial. We may indeed say that we call the yelocities in one configuration the same as those in another when the quantities gi, "-gn have the same values in the two cases. But this signifies nothing until the system of coordinates has been defined. We might identify the velocities in the two cases which make the quantities Pif'Pn the same in each. This again would signify nothing independently of the system of coordinates employed. 62 EXTENSION IN CONFIGURATION within certain infinitesimal limits of velocity. The second of these numbers divided by the first expresses the probability that a system which is only specified as falling within the in- finitesimal limits of configuration shall also fall within the infinitesimal limits of velocity. If the limits with respect to velocity are expressed by the condition that the momenta shall fall between the limits p-^ and p^ + dp^, . . .p^ and P% + dpn^ tiie extension-in-velocity within those limits will be Ap*cfpi . . . dp^f and we may express the probability in question by e^^^dp^ . . . dp^. (162) This may be regarded as defining rjp. The probability that a system which is only specified as having a configuration within certain infinitesimal limits shall also fall within any given limits of velocity will be expressed by the multiple integral f...Je'^\idp^...dp^, (163) or its equivalent J. . .Je'^^^^dq, . . . dq,, (164) taken within the given limits. It follows that the probability that the system will fall within the limits of velocity, q^ and q^ + dq^y . . . gw and ?• + ^« is expressed by e"^ ^ii dq^ . . . di^. (165) The value of the integrals (163), (164) is independent of the system of cot5rdinates and momenta which is used, as is also the value of the same integrals without the factor e'*; therefore the value of rj^ must be independent of the system of coordinates and momenta. We may call e'p the coefficient of prohabUity of velocity, and rjp the iridex of proba- hility of velocity* AND EXTENSION IN VELOCITY. 63 Comparing (160) and (162) with (40), we get e'^e'p = P = e'' (166) or Vq + Vp = V' (167) That is : the product of the coefficients of probability of con- figuration and of velocity is equal to the coefficient of proba- bility of phase; the sum of the indices of probability of configuration and of velocity is equal to the index of probability of phase. It is evident that e'« and e'p have the dimensions of the reciprocals of extension-in-configuration and extension-in- velocity respectively, i. e., the dimensions of ir^ €~5 ahd €~», where t represent any time, and e any energy. If, therefore, the unit of time is multiplied by c<, and the unit of energy by c,, every r}g will be increased by the addition of w log c, + Jn log c., (168) and every Vp by the addition of in log c,* (169) It should be observed that the quantities which have been called extension-in-configuration and extenBion-in-vdocity are not, as the terms might seem to imply, purely geometrical or kinematical conceptions. To express their nature more fully, they might appropriately have been called, respectively, the dynamical meamire of the extension in configuration^ and the dynamical measure of the extension in velocity. They depend upon the masses, although not upon the forces of the system. In the simple case of material points, where each point is limited to a given space, the extension-in-configuration is the product of the volumes within which the several points are confined (these may be the same or different), multiplied by the square root of the cube of the product of the masses of the several points. The extension-in-velocity for such systems is most easily defined as the extension-in-configuration of systems which have moved from the same configuration for the unit of time with the given velocities. * Compare (47) in Chapter L 64 EXTENSION IN CONFIGURATION In the general case, the notions of extension-in-configuration and extension-in-velocity may be connected as follows. If an ensemble of similar systems of n degrees of freedom have the same configuration at a given instant, but are distrib- uted throughout any finite extension-in-velocity, the same ensemble after an infinitesimal interval of time ht will be distributed throughout an extension in configuration equal to its original extension-in-velocity multipUed by S^-. In demonstrating this theorem, we shall write y^', . . . §'„' for the initial values of the coordinates. The final values will evidently be connected with the initial by the equations ^1 — fi'i' = i^^> "-qn — qn'^ qu^' (170) Now the original extension-in-velocity is by definition repre- sented by the integral J. . .J^iUq, . . . di^, (171) where the limits may be expressed by an equation of the form ^{3u'*'qn) = 0. (172) The same integral multiplied by the constant Sf^ may be written r. . .fAi^d(qiSt), . . . d{qjt)y (173) and the limits may be written ^(ii . . . i) =/(gi«, . . . g.Se) =0. (174) (It will be observed that Bt as well as Aj is constant in the integrations.) Now this integral is identically equal to f. . .J\^ rf(gi - giO ...d{q^... qj), (175) or its equivalent J^'^Jl^'^dq^...dq^, (176) with limits expressed by the equation J (qi - qi'f . . . s'. - qJ) = o. (177) AND EXTENSION IN VELOCITY. 65 But the systems which initially had velocities satisfying the equation (172) will after the interval ht have configurations satisfying equation (177). Therefore the extension-in-con- figuration represented by the last integral is that which belongs to the systems which originally had the extension-in- velociiy represented by the integral (171). Since the quantities which we have called extensions-in- phase, extensions-in-configuration, and extensions-in-velocity are independent of the nature of the system of coordinates used in their definitions, it is natural to seek definitions which shall be independent of the use of any codrdinates. It wUl be sufficient to give the following definitions without formal proof of their equivalence with those given above, since they are less convenient for use than those founded on systems of co- ordinates, and since we shall in fact have no occasion to use them. We commence with the definition of extension-in-velocity. We may imagine n independent velocities, F^ , . . . F], of which a system in a given configuration is capable. We may conceive of the system as having a certain velocity V^ combined with a part of each of these velocities V-^.^.V,^. By a part of F^ is meant a velocity of the same nature as V-^ but in amount being anything between zero and 1^. Now all the velocities* which may be thus described may be regarded as forming or lying in a certain extension of which we desire a measure. The case is greatly simplified if we suppose that certain relations exist between the velocities 1^ , . . . F^, viz : that the kinetic energy due to any two of these velocities combined is the sum of the kinetic energies due to the velocities separately. In this case the extension-in-motion is the square root of the product of the doubled kinetic energies due to the n velocities P^ , . . . V^ taken separately. The more general case may be reduced to this simpler case as follows. The velocity Kj may always be regarded as composed of two velocities V^ and F^", of which V^ is of the same nature as V^ , (it may be more or less in amount, or opposite in sign,) while F^" satisfies the relation that the 6 66 EXTENSION IN CONFIGURATION kinetic energy due to V\ and F^" combined is the sum of the kinetic energies due to these velocities taken separately. And the velocity V^ may be regarded as compounded of three, ^z^ K"y Vz"^ of which Fg' is of the same nature as V^ , V^' of the same nature as V^\ while V^" satisfies the relations that if combined either with V-^ or V^' the kinetic energy of the combined velocities is the sum of the kinetic energies of the velocities taken separately. When all the velocities F^ , . . . Vn have been thus decomposed, the square root of the product of the doubled kinetic energies of the several velocities Fj, Fg", V^"j etc., will be the value of the extension-in- velocity which is sought. This method of evaluation of the extension-in- velocity which we are considering is perhaps the most simple and natural, but the result may be expressed in a more symmetrical form. Let us write 6^2 for the kinetic energy of the velocities F^ and V^ combined, diminished by the sum of the kinetic energies due to the same velocities taken separately. This may be called the mutual energy of the velocities V^ and V^. Let the mutual energy of every pair of the velocities F^ , . . . F^ be expressed in the same way. Analogy would make e^^ represent the energy of twice F^ diminished by twice the energy of F^ , i. e.y e^i would represent twice the energy of V^ , although the term mutual energy is hardly appropriate to this case. At all events, let e^ have this signification, and €^^ represent twice the energy of F^, etc. The square root of the determinant €11 €12 . . . €lf^ €ai €22 . . . €211 represents the value of the extension-in-velocity determined as above described by the velocities 1^ , . . . F],. The statements of the preceding paragraph may be readily proved from the expression (167) on page 60, viz., by which the notion of an element of extension-in-velocity was AND EXTENSION IN VELOCITY. 67 originally defined. Since Jj in this expression represents the determinant of which the general element is dq\dqj the square of the preceding expression represents the determi- nant of which the general element is Now we may regard the differentials of velocity rfj^, dqj as themselves infinitesimal velocities. Then the last expression represents the mutual energy of these velocities, and dh . represents twice the energy due to the velocity dqt . The case which we have considered is an extension-in-veloc- ity of the simplest form. All extensions-in-velocity do not have this form, but all may be regarded as composed of elementary extensions of this form, in the same manner as all volumes may be regarded as composed of elementary parallelepipeds. Having thus a measure of extension-in- velocity founded, it will be observed, on the dynamical notion of kinetic energy, and not involving an explicit mention of coordinates, we may derive from it a measure of extension-in-configuration by the principle connecting these quantities which has been given in a preceding paragraph of this chapter. The measure of extension-in-phase may be obtained from that of extension-in-configuration and of extension-in-velocity. For to every configuration in an extension-in-phase there will belong a certain extension-in-velocity, and the integral of the elements of extension-in-configuration within any extension- in-phase multiplied each by its extension-in-velocity is the measure of the extension-in-phase. CHAPTER Vn. FARTHER DISCUSSION OF AVERAGES IN A CANONICAL ENSEMBLE OF SYSTEMS. Returning to the case of a canonical distribution, we have for the index of probability of configuration V, = ^ (178) as appears on comparison of formulae (142) and (161). It follows immediately from (142) that the average value in the ensemble of any quantity u which depends on the configura- tion alone is given by the formula u = I .. . j ue ® A^dqi...dq^j (179) oonfig. where the integrations cover all possible configurations. The value of yftg is evidently determined by the equation e ® =/.../ ^A^^dqi . . . dq^. (180) oonfig. By differentiating the last equation we may obtain results analogous to those obtained in Chapter IV from the equation e ^ = j . , . j e ^dpi . . . dq^^. phaaes As the process is identical, it is sufficient to give the results : dijfg = rj^d® — J[idai — J^da^ — etc., (181) AVERAGES IN A CANONICAL ENSEMBLE. 69 or, since ^^ = 7, + 0^,, (182) and d^^ = rf^ + Vq^® + ®d^«, (183) d€q = — ©d^g — 2idai — A^da^ — etc. 0-^) It appears from this equation that the differential relations subsisting between the average potential energy in an ensem- ble of systems canonically distributed, the modulus of distri- bution, the average index of probability of configuration, taken negatively, and the average forces exerted on external bodies, are equivalent to those enunciated by Clausius for the potential energy of a body, its temperature, a quantity which he called the disgregation, and the forces exerted on external bodies.* For the index of probability of velocity, in the case of ca- nonical distribution, we have by comparison of (144) and (163), or of (145) and (164), which gives Vp = ^ ^^ ; (186) we have also ^ = ^ n 0, (187) and by (140), i/r^ = - ^ n log (27r0). (188) From these equations we get by differentiation d^, = Vpd®y (189) and ct, = — rfv (190) The differential relation expressed in this equation between the average kinetic energy, the modulus, and the average index of probability of velocity, taken negatively, is identical with that given by Clausius locia citatis for the kinetic energy of a body, the temperature, and a quantity which he called the transformation-value of the kinetic energy, f The relations € = €g + €„ 17 = ^^ + 17p ♦ Pogg. Ann., Bd. CXVI, S. 73, (1862) ; ibid., Bd. CXXV, S. 863, (1865). See also Boltzmann, Sitzb. der Wiener Akad., Bd. LXIII, S. 728, (1871). t YerwandlungBwerth des W&rmeinhaltes. 70 AVERAGE VALUES IN A CANONICAL are also identical with those given by Clausius for the cone- spending quantities. Equations (112) and (181) show that if -^ or -^^ is known as function of 6 and a^, a^, etc., we can obtain by differentia- tion € or €^ and Ai, A^, etc. as functions of the same varia- bles. We have in fact 7=^ — ®^=,/r — 0^. (191) J f The corresponding equation relating to kinetic energy, ^ = l^,-017, = l^^-0^^ (193) which may be obtained in the same way, may be verified by the known relations (186), (187), and (188) between the variables. We have also etc., so that the average values of the external forces may be derived alike from ^fr or from yjrg. The average values of the squares or higher powers of the energies (total, potential, or kinetic) may easily be obtained by repeated differentiations of '^, ^frgy yjtp^ or e, ig, e,, with respect to ©. By equation (108) we have /all y— € . . . j €e ® dpi. . . dq^, (196) aU e phaaei and differentiating with respect to ©, S=/-/(=^+iSF'^-*- <'»«' d® phases whence, again by (108), ENSEMBLE OF SYSTEMS. 71 ? = 0«| + i(^-®|). (197) Combining this with (191), €-* = € + &§,= (^- ©^Y- 0«g. (198) d® y dQ>J d/^ ^ ^ In precisely the same way, from the equation €« = (...( €^e ® Ai^ dqi . . . dq^, (199) config. we may obtain p *« =?+»■§=(*.-« t)'-«-^'- « In the same way also, if we confine ourselves to a particular configuration, from the equation ^P TClOO. we obtain =J...Je,e » C^}dpi...dp„ (201) V ?+«'S=(*.-«^)-«'^' (^) which by (187) reduces to ^=(Jn2 + Jn)®2. (203) Since this value is independent of the configuration, we see that the average square of the kinetic energy for every configu- ration is the same, and therefore the same as for the whole ensemble. Hence e^ may be interpreted as the average either for any particular configuration, or for the whole ensemble. It will be observed that the value of this quantity is deter- mined entirely by the modulus and the number of degrees of freedom of the system, and is in other respects independent of the nature of the system. Of especial importance are the anomalies of the energies, or their deviations from their average values. The average value 72 AVERAGE VALUES IN A CANONICAL of these anomalies is of course zero. The natural measure of such anomalies is the square root of their average square. Now G-;^)^ = ?-r, (204) identically. Accordingly (^^=®'S = -®'0- (205) In like manner, (^ -7y = ©«^ = -0»^, (206) (^-^)* = ®"^ = -®'^^ = t^®"- (207) ^p d® d0* Hence (c - -c)« = (c, - l,y + (^ - V'- (208) Equation (206) shows that the value of dCeJdB can never be negative, and that the value of d^yjrg/d&^ or drjg/dS can never be positive.* To get an idea of the order of magnitude of these quantities, we may use the average kinetic energy as a term of comparison, this quantity being independent of the arbitrary constant in- volved in the definition of the potential energy. Since * In the case discussed in the note on page 64, in which the potential energy is a quadratic function of the q*8, and A^ independent of the ^'s, we should get for the potential energy and for the total energy We may also write in this case, (e -€)« _ 1 ENSEMBLE OF SYSTEMS. 78 € =• Jn®, (209) (210) (1Z15)! = ?^ = ? + H^^. (211) These equations show that when the number of degrees of freedom of the systems is very great, the mean squares of the anomalies of the energies (total, potential, and kinetic) are very small in comparison with the mean square of the kinetic energy, unless indeed the differential coefficient dejdep is of the same order of magnitude as n. Such values of dejdip can only occur within intervals QJ* — e^') which are of the or- der of magnitude of ti""^, unless it be in cases in which e^ is in general of an order of magnitude higher than ^. Postponing for the moment the consideration of such cases, it will be in- teresting to examine more closely the case of large values of dcg/dcp within narrow limits. Let us suppose that for ^' and ip" the value of dcg/dcp is of the order of magnitude of unity, but between these values of ip very great values of the differ- ential coefficient occur. Then in the ensemble having modulus 0" and average energies e^" and e^", values of e^ sensibly greater than €g" will be so rare that we may call them practically neg- ligible. They will be still more rare in an ensemble of less modulus. For if we differentiate the equation regarding €g as constant, but 6 and therefore yjtg as variable, we get /^A l#,_t/r,-^6 2 . \d®J,^ ® d® ®» ' ^ ^ whence by (192) (213) (: ^Vq \ _ €g — €g rf0A. 0^ 74 AVERAGE VALUES IN A CANONICAL That is, a diminution of tiie modulus will diminish tiie proba- bility of all configurations for which the potential energy exceeds its average value in the ensemble. Again, in the ensemble having modulus ©' and average energies e,,' and e^', values of €g sensibly less than ig' will be so rare as to be practically neg- ligible. They will be still more rare in an ensemble of greater modulus, since by the same equation an increase of the modulus will diminish the probability of configurations for which the potential energy is less than its average value in the ensemble. Therefore, for values of © between & and ©", and of €p between €p' and tp", the individual values of e^ will be practically limited to the interval between e^' and e^ '. In the cases which remain to be considered, \dz., when dejdep has very large values not confined to narrow limits, and consequentiy the differences of the mean potential ener- gies in ensembles of different moduli are in general very large compared with tiie differences of the mean kinetic energies, it appears by (210) that the anomalies of mean square of poten- tial energy, if not small in comparison witii the mean kinetic energy, wiU yet in general be very small in comparison with differences of mean potential energy in ensembles having moderate differences of mean kinetic energy, — the exceptions being of the same character as described for the case when d€q/d€p is not in general large. It follows that to human experience and observation with respect to such an ensemble as we are considering, or witii respect to systems which may be regarded as taken at random from such an ensemble, when the number of degrees of free- dom is of such Older of magnitude as tiie number of molecules in the bodies subject to our observation and experiment, e — e, €,-€„ €g-€q would be m general vanishing quantities, since such experience would not be wide enough to embrace the more considerable divergencies from the mean values, and such observation not nice enough to distinguish tiie ordinaiy divergencies. In other words, such ensembles would appear to human observation as ensembles of systems of uniform energy, and in which the potential and kinetic energies (sup- ENSEMBLE OF SYSTEMS. 75 posing that there were means of measuring these quantities separately) had each separately uniform values.* Exceptions might occur when for particular values of the modulus the differential coefficient dcg/cCep takes a very large value. To human observation the effect would be, that in ensembles in which 6 and e, had certain critical values, Cg would be in- determinate within certain limits, viz., the values which would correspond to values of © and Cp slightly less and slightly greater than the critical values. Such indeterminateness cor- responds precisely to what we observe in experiments on the bodies which nature presents to us.t To obtain general formulae for the average values of powers of the energies, we may proceed as follows. If A is any posi- tive whole number, we have identically aU ^f^ aU _^ J . . . j £»e~»dpx ...dqn= ®'-^f- • •/«*"'« ®<^i'i •••<*?•' (214) phaaes phaaea t. e., by (108), ?e ® = ®'^(«^6 *Y (215) Hence »_i ^' * = ifli) " *' <216) ♦ . -»_* and ? = 6®/^®»^^e® (217) * This impUes that the kinetic and potential energies of individual Bystemg would each separately have values sensibly constant in time. t As an example, we may take a system consisting of a fluid in a cylinder under a weighted piston, with a vacuum between the piston and the top of the cylinder, which is closed. The weighted piston is to be regarded as a part of the system. (This is formaUy necessary in order to satisfy the con- dition of the invariability of the external coordinates.) It is evident that at a certain temperature, viz., when the pressure of saturated vapor balances the weight of the piston, there is an indeterminateness in the values of the potential and total energies as functions of the temperature. 76 AVERAGE VALUES IN A CANONICAL For A = 1, this gives ^ = -^•^(1) (218) which agrees with (191). From (216) we have also ? = ^^+®'5=(^+®'^)^*. (219) (J V A 1 In like maimer from the identical equation oonfig. oonfig. (221) we get ^ = '®V®'rf^)^ ®' (222) and €7 = •« (^« + ®'l)"^«- (223) With respect to the kinetic energy similar equations wiU hold for averages taken for any particular configuration, or for the whole ensemble. But since the equation d V (^P + ®'^)^ (224) reduces to ^=(i®+'®^)^=is®+®'^r®- ^"^^ C/=(x ENSEMBLE OF SYSTEMS. 77 We have therefore ^=g + l)^©*. (226) = (l + =^)(i + ^)I^- <227) The average values of the powers of the anomalies of the energies are perhaps most easily found as follows. We have identically, since e is a function of ©, while e is a function of the ^'s and y's, J*..-J [e(€-i)*~A(€-?)»-i0«^Je ^dp^,...dq„ (229) phaiifti i. e., by (108), ®'^[("^^«~'*J = [e («-«)* - A (€--£)»-» ©"^J r®, (230) * In the case discussed in the note on page 54 we may easily get which, with «g — €a = s ®> gives Hence (€*i + 10Z>€2>^ (€ - €)• = 2>«c + l^D'elTe + 10 (D^i)^ + 15 {Dlf etc. where 2> represents the operator &^d/dS. Similar expres- sions relating to the potential energy may be derived from (232). For the kinetic energy we may write similar equations in which the averages may be taken either for a single configura- tion or for the whole ensemble. But since dcp n 5© "2 the general formula reduces to (ep-ip)^^ = ©^^(€p-ip)* + inA0«(6^-€p)" (233) or V*-l (gp - €p) ^^ -- ?® _1 (gp - gj>) * . 2A (^p^^JpI* + ^ (€p - epY (234) ^»+i n d® ip» n e/ « ep*-^ ENSEMBLE OF SYSTEMS. 79 But since identically (6p - "cpY ^ ^^ (g - ef ^ Q^ the value of the corresponding expression for any index will be independent of © and the formula reduces to i^r -"{-^y ^"K^r <-> Cj, X ^ ^P we have therefore (^y=^' (^-y^. (^y=°' i^y-^-'^- i-^y-i etc.* It wiU be observed that when -^ or i is given as function of ©, all averages of the form ? or (e — e)* are thereby deter- * In the case discussed in the preceding foot-notes we get easily (€g-e,)»=(«p-ep)». — M^\h y« ,'v» and ('AI1I1\-^('X^\' \€^— €a^ \ €p / For the total energy we have in this case /«^\a 1 /jLZ-iV-l .1 V7-J =n- Vc-J -»»^ + ««' ( J~* I =_, etc. 80 AVERAGE VALUES IN A CANONICAL mined* So also if yjr^ or €^ is giyen as function of B, all averages of the form 7^ or (e^ — €^)* are determined. But Therefore if any one of the quantities -^^ y^gy i, e^ is known as function of B, and n is also known, all averages of any of the forms mentioned are thereby determined as functions of the same variable. In any case all averages of the form are known in terms of n alone, and have the same value whether taken for the whole ensemble or limited to any particular configuration. If we differentiate the equation J '. .1 e^ dpi...dq^ = l (236) plUUMB with respect to a^, and multiply by ©, we have Differentiating again, with respect to a^, with respect to a^^ and with respect to ©, we have J J \^daida2 daida2 ®\dai daij\da2 dOj/J e^ dpi...dqn = Oj (239) J **J \jdaid% \(foi daij \®d® ®* /J 6 ® i^i . . . d^, = 0. (240) ENSEMBLE OF SYSTEMS. 81 The multiple integrals in the last four equations represent the average values of the expressions in the brackets, which we may therefore set equal to zero. The first gives ^ = g = -J., (241, as already obtained. With this relation and (191) we get from the other equations =0f^_f:^^=0f§^_^^ (243) \dat datj \<*cii aa-ij We may add for comparison equation (205), which might be derived from (236) by differentiating twice with respect to © : '(^^r = -^2-''W (244) The two last equations give dA- (A - ^0(^ - ^ = -^(^ - ^y. (246) de If -^ or € is known as function of ©, a^, a^^ etc., (e — e)^ may be obtained by differentiation as function of the same variables. And if yjr^ or -J^, or ^ is known as function of 0, a^, etc., ^1 — A{) (e — €) may be obtained by differentiation. But (-^1 — A{)^ and (-4^ — 2.{) (-^3 — Jfj) cannot be obtained in any similar manner. We have seen that (e— e)^ is in general a vanishing quantity for very great values of n^ which we may regard as contained implicitly in © as a divisor. The same is true of (A^ — A^^ (c — - i). It does not appear that we can assert the same of (A^ — A{)^ or (^A^ — A{) (^A^ — ^2), since 6 82 AVERAGE VALUES IN A CANONICAL dJ^eJda^ may be very great. The quantities dhfda^ and cPyjrlda^ belong to the class called elasticities. The former expression represents an elasticity measured under the condition that while a^ is varied the internal cob'rdinates j^, ... g'^ all remain fixed. The latter is an elasticity measured under the condi- tion that when a^ is varied the ensemble remains canonically distributed within the same modulus. This corresponds to an elasticity in physics measured under the condition of con- stant temperature. It is evident that the former is greater than the latter, and it may be enormously greater. The divergences of the force A-^ from its average value are due in part to the differences of energy in the systems of the ensemble, and in part to the differences in the value of the forces which exist in systems of the same energy. If we write -27]g for the average value of -4 ^ in systems of the ensemble which have any same energy, it wiU be determmed by the equation 211. = -^ -^ ^ (246) /•/■ dpi . . . dg^ where the limits of integration in both multiple integrals are two values of the energy which differ infinitely little, say e and € + de. This wiU make the factor e ® constant within the limits of integration, and it may be cancelled in the numera- tor and denominator, leaving I ... I — ;^dpi . . . e^„ 2^, = '^ / P (247) where the integrals as before are to be taken between € and € + de, 3^ is therefore independent of 0, being a function of the energy and the external cot)rdinates. ENSEMBLE OF SYSTEMS. 88 Now we have identically where -4^ — 33« denotes the excess of the force (tending to increase a^) exerted by any system above the average of such forces for systems of the same energy. Accordingly, But the average value of (-4^ — ^Tlt) (-271 e — -^j) for systems of the ensemble which have the same energy is zero, since for such systems the second factor is constant. Therefore the average for the whole ensemble is zero, and {A, - ^0' = (^1 - '^liY + (^1 ^ - ^i)'. (248) In the same way it may be shown that (A-J[0(6-^ = (^.-Zi)(€-^. (249) It is evident that in ensembles in which the anomalies of energy € — € may be regarded as insensible the same will be true of the quantities represented by 371* "" ^v The properties of quantities of the form 3^ will be farther considered in Chapter X, which will be devoted to ensembles of constant energy. It may not be without interest to consider some general formulae relating to averages in a canonical ensemble, which embrace many of the residts which have been given in this chapter. Let u be any function of the internal and external coordi- nates witii the momenta and modulus. We have by definition ... I ue^ dpi. . .dq^ (250) pbaaei If we differentiate with respect to ©, we have phfwef 84 AVERAGE VALUES IN A CANONICAL Setting u = 1 in this equation, we get dij; ij/ — € and substituting this value, we have du ^du U€ U€ du ^^du — If we differentiate equation (250) with respect to a (which may represent any of the external coordinates), and write A for the force — j-j > we get du __ r r[ du u di// u \ d^''J"'J\d^®di^ ^® J e ® dpi • • * dq^^ phases du du u d\b . uA /ft^ov or — = ^H (263) da da^ %da^ % ^ ' Setting t^ = 1 in this equation, we get Substituting this value, we have du ^c[u U~K uA fOKA\ da da ® % * ^ ^ du du — or ®2 ®-^ = u2.'-ul=(u — u)(A--A). (255) Repeated applications of the principles expressed by equa- tions (252) and (255) are perhaps best made in the particular cases. Yet we may write (252) in this form ENSEMBLE OF SYSTEMS. 85 (e + i>) (w - t^) = 0, (256) where D represents the operator ©^ d/d©. Hence (e + Df (w - ii) = 0, (257) where A is any positive whole number. It will be observed, that since e is not function of ©, (e + jD)* may be expanded by the binomial theorem. Or, we may write (e + D) « = (e + D) w, (258) whence (e + i>)* w = (i + Df u. (259) But the operator (e + i>)*, although in some respects more simple than the operator without the average sign on the €, cannot be expanded by the binomial theorem, since € is a function of with the external coordinates. So from equation (254) we have ( ¥ + 1) («-«) = «' <2«>) whence (^ + ^)* ('^ - «> = ^ 5 (2^^) The binomial theorem cannot be applied to these operators. Again, if we now distinguish, as usual, the several external coordinates by sufl&xes, we may apply successively to the expression u — u any or all of the operators 86 AVERAGES IN A CANONICAL ENSEMBLE. as many times as we choose, and in any order, the average value of the result will be zero. Or, if we apply the same operators to u, and finally take the average value, it will be the same as the value obtained by writing the sign of average separately as u, and on e, A^j A^y etc., in all the operators. If u is independent of the momenta, formulae similar to the preceding, but having eg in place of €| may be derived from equation (179). CHAPTER Vm. ON CERTAIN IMPORTANT FUNCTIONS OF THE ENERGIES OF A SYSTEM. In Older to consider more particiilarly the distribution of a canonical ensemble in energy, and for other purposes, it will be convenient to use the following definitions and notations. Let us denote by Fthe extension-in-phase below a certain limit of energy which we shaU. call 6. That is, let F= i. . . jdpi . . . dq^y (265) the integration being extended (with constant values of the external coordinates) over all phases for which the energy is less than the limit e. We shall suppose that the value of this integral is not infinite, except for an infinite value of the lim- iting energy. This will not exclude any kind of system to which the canonical distribution is applicable. For if /••/ e ® dpi . . . dq^ taken without limits has a finite value,* the less value reprer sented by 6 € e J .. .J dpi.. .dq^ taken below a limiting value of €, and with the e before the integral sign representing that limiting value, will also be finite. Therefore the value of FJ which differs only by a constant factor, will also be finite, for finite e. It is a func- tion of 6 and the external coordinates, a continuous increasing * This IB a necessary condition of the canonical distribution. See Chapter IV, p. 85. 88 CERTAIN IMPORTANT FUNCTIONS function of €, which becomes infinite with e, and vanishes for the smallest possible value of c, or f or c = — oo, if the energy may be diminished without limit. Let us also set * = log ^. (266) The extension in phase between any two limits of energy, ^ and e", will be represented by the integral X * e* de. (267) £ And in general, we may substitute €* de for dp^ . . . dq^ in a 2n-fold integral, reducing it to a simple integral, whenever the limits can be expressed by the energy alone, and the other factor under the integral sign is a function of the energy alone, or with quantities which are constant in the integration. In particular we observe that the probability that the energy of an unspecified system of a canonical ensemble lies between the limits e' and e" will be represented by the integral * f^e^^^de, (268) and that the average value in the ensemble of any quantity which only varies with the energy is given by the equation f we ® de, (269) where we may regard the constant -^ as determined by the equation J e ®=/ e ^ de, (270) In regard to the lower limit in these integrals, it will be ob- served that F= is equivalent to the condition that the value of 6 is the least possible. * Compare equation (93). t Compare equation (108). I Compare equation (02). OF THE ENERGIES OF A SYSTEM. 89 In like manner, let us denote by F, the extension-in-configu- ration below a certain limit of potential energy whicli we may call €g. That is, let V, = r. . .J^^dq^ ...dq,, (271) the integration being extended (with constant values of the external coordinates) over all configurations for which the potential energy is less than e^. Vg will be a function of e^ with the external coordinates, an increasing function of e^, which does not become infinite (in such cases as we shall con- sider *) for any finite value of 6^ It vanishes for the least possible value of e^, or for 6^ = — oo , if e^ can be diminished without limit. It is not always a continuous function of e^. In fa<5t, if there is a finite extension-m-configuration of con- stant potential energy, the corresponding value of Vg will not include that extension-in-configuration, but if Cg be in- creased infinitesimally, the corresponding value of Vg will be increased by that finite extension-in-configuration. Let us also set *' = log ^'- (272) The extension-in-configuration between any two limits of potential energy €g and e/' may be represented by the integral r e^'^deg (273) '«' whenever there is no discontinuity in the value of F^ as function of e^ between or at those limits, that is, when- ever there is no finite extension-in-configuration of constant potential energy between or at the limits. And in general, with the restriction mentioned, we may substitute e^« rfe, for A^ rfjx • • • ^?» ^^ ^^ w-f old integral, reducing it to a simple integral, when the limits are expressed by the potential energy, and the other factor under the integral sign is a function of *llVq were infinite for finite values of c,, F would evidently be infinite for finite values of c 90 CERTAIN IMPORTANT FUNCTIONS the potential energy, either alone or with quantities which are constant in the integration. We may often avoid the inconvenience occasioned by for- mulae becoming iUusoiy on account of discontinuities in the values of V^ as function of e^ by substituting for the given discontinuous function a conthiuous function which is practi- cally equivalent to the given function for the purposes of the evaluations desired. It only requires infinitesimal changes of potential energy to destroy the finite extensions-in-configura- tion of constant potential energy which are the cause of the difficulty. In the case of an ensemble of systems canonically distributed in configuration, when V^ is, or may be regarded as, a continu- ous function of e^ (within the limits considered), the prober bility that the potential energy of an unspecified system lies between the limits e/ and e^" is given by the integral / ^''^■^^. 6 « cfe,, (274) where -^ may be determined by the condition that the value of the integral is unity, when the limits include all possible values of e^ In the same case, the average value in the en- semble of any function of the potential energy is given by the equation ra=o e ® cfc,. (275) When Vq is not a continuous function of e^, we may write dVq for ^^d€g in these formulae. In like manner also, for any given configuration, let us denote by V^ the extension-in-velocity below a certain limit of kinetic energy specified by e^. That is, let Fp = r. . . fAp* rfi)i • . • dp,, (276) OF THE ENERGIES OF A SYSTEM. 91 the integration being extended, with constant values of the coordinates, both internal and external, over all values of the momenta for which the kinetic energy is less than the limit e^. Vp will evidently be a continuous increasing function of €p which vanishes and becomes infinite with ep. Let us set The extension-in-velocity between any two limits of kinetic energy ep' and 6," may be represented by the integral / V e'^'^ckp. (278) And in general, we may substitute e^^ de^ for A,* dp^ . . . dp^ or Ag* dq^ . . . dq^ in an w-fold integral in which the coordi- nates are constant, reducing it to a simple integral, when the limits are expressed by the kinetic energy, and the other factor imder the integral sign is a function of the kinetic energy, either alone or with quantities which are constant in the integration. It is easy to express Vp and ^p in terms of e,. Since Ap is function of the coordinates alone, we have by definition r,=^\*J...Jdp,...dp^ (279) the limits of the integral being given by Cp. That is, if €p=^F(pu...Pn), (280) the limits of the integral for e, = 1, are given by the equation i^(i>i,...i>n) = l, (281) and the limits of the integral for e, = a\ are given by the equation F(pt,...pn)=a\ (282) But since F represents a quadratic function, this equation may be written 92 CERTAIN IMPORTANT FUNCTIONS The value of Vp may also be put in the f onn Vp = a*d^ f... fd^ • • • ^~ • (284) Now we may determine Vp for «, = 1 from (279) where the limits are expressed by (281), and V^ for c, = a^ from (284) taking the limits from (283). The two integrals thus deter- mined are evidently identical, and we have (^^W = «"(^|.W (286) i. e., Vp varies as e/. We may therefore set n n V,= Ce,^ e'^lOe,^ (286) 2 where C is a constant, at least for fixed values of the internal cotirdinates. To determine this constant, let us consider the case of a canonical distribution, for which we have 6 ® d€p = l, £' '0 where e^ = (2w®) 2. Substituting this value, and that of e*" from (286), we get ^^x 00 _5? «-_! « 6 ®€/ rf€p = (27r®)2, «o n ^Cr(|) = (2.A (287) r(i« + 1) Having thus detemuned the value of the constant 0, we may OF THE ENERGIES OF A SYSTEM. 98 substitute it in the general expressions (286), and obtain the following values, which are perfectly general: n .2 (288) n n_- i2.2 It will be observed that the values of V^ and ^p for any given €j, are independent of the configuration, and even of the nature of the system considered, except with respect to its number of degrees of freedom. Returning to the canonical ensemble, we may express the probability that the kinetic energy of a system of a given configuration, but otherwise unspecified, falls within given limits, by either member of the following equation Since this value is independent of the cob'rdinates it also represents the probability that the kinetic energy of an unspecified system of a canonical ensemble falls within the limits. The form of the last integral also shows that the prob- ability that the ratio of the kinetic energy to the modulus • Very similar values for Vq^ e^ff, F, and e^ may be found in the same way in the case discussed in the preceding foot-notes (see pages 54, 72, 77, and 79), in which e^ is a quadratic function of the q*%, and A^ independent of the ^'s. In this case we have -(^)* {2»)5(,,-,„)5 r(in + l) • 9' n Va,/ r(n + l) • ■■t_/^\* (2ir)"(«- ».)""' 94 CERTAIN IMPORTANT FUNCTIONS ialls within given limits is independent also of the Talne of the modulus, being detennined entiiely fay the number of degrees of freedom of the system and the limiting values of the ratio. The average value of any function of the kinetic eneigy, either for the whole ensemble, or for any particular configura- tion, is given fay u=—^ / ue ^€,* d€, •(291) e'ran) Thus: — ^r(m+|n)^^ ^ m + iit>0; t(292) T(in) n * The corresponding equation for the arerage raloe of anj function of the potential energy, when this is a quadratic function of the ^s, and A^ is independent of the q'», is « = -- /ue ® («« — «a)' d€q. In the same case, the arerage ralue of anj function of the (total) energy is giren by the equation Hence in this case (ir^Tj= = ?l^j±i^ e-. if m + ln>0. and g = l. if »>1. If n = 1, 6* = 2ir and rf^/rff = for any value of €. If n = 2, the case is the same with respect to i>^ t This equation has already been proved for positive integral powers of the kinetic energy. See page 77. OF THE ENERGIES OF A SYSTEM. 95 n n '*' = ^j;^(2t)202 ^ if „>i. (294) ^ = 1, if »>2; (296) e"^'' F; = a (296) If w = 2, 6*^ = 2 TT, and dp/d€j, = 0, for any value of 6^. The definitions of FJ F^ and Vp give r^JJdVpdVg (297) where the integrations cover all phases for which the energy is less than the limit 6, for which the value of F^is sought This gives V=j VpdVg, (298) F«=0 /=-£=[ e*'dr,, (299) Vq=0 where Vp and c^ are connected with Vg by the equation €p + €q = constant = €• (300) If w > 2, €^ vanishes at the upper limit, i. e., for ^ = 0, and we get by another differentiation Fg=0 We may also write F= r Vpe^^de^, (302) Ffl=0 / = r 6*^"^*«cfc„ (303) Ft=0 96 CERTAIN IMPORTANT FUNCTIONS eto^ when F^ is a continuous function of Cg commencing with the value F^ = 0, or when we choose to attribute to F^ a fictitious continuity commencing with the value zero, as de- scribed on page 90. If we substitute in these equations the values of V, and c^ which we have found, we get ^=r-(iST^/(*-^)'''^- (304) n _ F«=0 where e^« de^ may be substituted for d F^ in the cases above described. If, therefore, n is known, and F^ as function of 6^ F^and €^ may be found by quadratures. It appears from these equations that F^is always a continu- ous increasing function of e, commencing with the value F"= 0, even when this is not the case with respect to F^ and e^. The same is true of e^, when n > 2, or when w = 2 if F^ in- creases continuously with €g from the value F^ = 0. The last equation may be derived from the preceding by differentiation with respect to 6. Successive differentiations give, if A < J w + 1, d^Vjd^ is therefore positive if A < J w + 1. It is an in- creasing function of e, if A < ^w. If e is not capable of being diminished without limit, cP^V/d^ vanishes for the least possible value of c, if A < Jw. If n is even, ^=(2»)i(F.),_, (307) OF THE ENERGIES OF A SYSTEM. 97 n That is, Vg is the same function of e^ as ; — — of €• When n is large, approximate formulae will be more avail- able. It will be sufficient to indicate the method proposed, without precise discussion of the limits of its applicability or of the degree of its approximation. For the value of e^ cor- responding to any given e, we have Cg = C c e* = y*e*''+*«de, = re*''+*« de,, (308) r,=o where tiie Tariables aie connected by the equation (300). The maximum value of ^p + ^^ is therefore characterized by the equation The values of €p and Cg determined by this maximum we shall distinguish by accents, and mark the corresponding values of functions of Cp and Cg in the same way. Now we have by Taylor's theorem *, = w + If the approximation is sufficient without going beyond the quadratic terms, since by (300) €p €p = — (€g — €q ), we may write e' = e'p / where the limits have been made ± oo for analytical simplicity. This is allowable when the quantity in the square brackets has a very large negative value, since the part of the integral 7 98 CERTAIN IMPORTANT FUNCTIONS corresponding to other than very small values of Cg — €g' may be regarded as a vanishing quantity. This gives - '' ' ' (313) or ^=v+*.'+iiog(2.).iiog[-(^y-.(^y]. (314) From this equation, with (289), (300) and (309), we may determine the value of ^ corresponding to any given value of €, when ^g is known as fimction of e^. Any two systems may be regarded as together forming a third system. If we have For ^ given as function of e for any two systems, we may express by quadratures Fand ^ for the system formed by combining the two. If we distinguish by the suffixes ( )ij C \^ ( )i2 ^^® quantities relating to the three systems, we have easily from the definitions of these quantities Fia = r fdVidV^ ^^CVadVi = CVidV^ = CVie^det, (315) e = Ce^dVi = Ce^d Va = fe^-^^'de^, (316) where the double integral is to be taken within the limits Fi = 0, Fi = 0, and Cj + Cj = c^, and the variables in the single integrals are connected by the last of these equations, while the limits are given by the first two, which characterize the least possible values of 6^ and 63 respectively. It will be observed that these equations are identical in form with those by which Fand ^ are derived from Vp or p and Vq or ^g, except that they do not admit in the general case those transformations which result from substituting for Vp or ^p the particular functions which these symbols always represent. OF THE ENERGIES OF A SYSTEM. 99 Similar formulae may be used to derive Vq or ^q for the compound system, when one of these quantities is known as function of the potential energy in each of the systems combined. The operation represented by such an equation as e = i e e d€t is identical with one of the fundamental operations of the theory of errors, viz., that of finding the probability of an error from the probabilities of partial errors of which it is made up. It admits a simple geometrical illustration. We may take a horizontal line as an axis of abscissas, and lay off €i as an abscissa measured to the right of any origin, and erect e^i as a corresponding ordinate, thus determining a certain curve. Again, taking a different origin, we may lay off €2 as abscissas measured to the left, and determine a second curve by erecting the ordinates g^a. We may suppose the distance be- tween the origins to be e^g' ^^® second origin being to the right if €12 is positive. We may determine a third curve by erecting at every point in the line (between the least values of ei and e^) an ordinate which represents the product of the two ordinates belonging to the curves already described. The area between this third curve and the axis of abscissas will represent the value of e^^. To get the value of this quantity for varying values of 612, we may suppose the first two curves to be rigidly con- structed, and to be capable of being moved independently. We may increase or diminish e^g by moving one of these curves to the right or left. The third curve must be constructed anew for each different value of e^g. CHAPTER IX. THE FUNCTION AND THE CANONICAL DISTRIBUTION. In this chapter we shall return to the consideration of the canonical distribution, in order to investigate those properties which are especially related to the function of the energy which we have denoted by ^. If we denote by JV, as usual, the total number of systems in the ensemble, will represent the nimiber having energies between the limits € and € + de. The expression ^,^+* (317) represents what may be called the density-in-energy. This vanishes for 6 = 00, for otherwise the necessary equation e^ de=zl (318) F=0 could not be fulfilled. For the same reason the density-in- energy will vanish f or e = — oo, if that is a possible value of the energy. Generally, however, the least possible value of the energy will be a finite value, for which, if w > 2, e* will vanish,* and therefore the density-in-energy. Now the density- in-energy is necessarily positive, and since it vanishes for extreme values of the energy Mn > 2, it must have a maxi- mum in such cases, in which the energy may be said to have * See page 06. THE FUNCTION «^. 101 its most common or most probable value, and which is detennined by the equation d^_ 1 de (319) This value of dffijde is also, when n > 2, its average value in the ensemble. For we have identically, by integration by parts. ft' .fe=|_e« J+i/<'* *. (320) F=0 V=0 V=0 If n > 2, the expression in the brackets, which multiplied by N would represent the density-in-energy, vanishes at the limits, and we have by (269) and (318) It appears, therefore, that for systems of more than two degrees of freedom, the average value of d^/de in an ensemble canoni- cally distributed is identical with the value of the same differ- ential coefficient as calculated for the most common energy in the ensemble, both values being reciprocals of the modulus. Hitherto, in our consideration of the quantities V] V^ Vp,y ^^ p9 ^^ have regarded the external coordinates as constant. It is evident, however, from their definitions that F^and ^ are in general f imctions of the external coordinates and the energy (e), that Vg and 0^ are in general functions of the external coordinates and the potential energy (e^). Vp and ^^ we have found to be functions of the kinetic energy (ep) alone. In the equation —21 /» —-+0 e ®=J e ® de, (322) v=o by which -^ may be determined, and the external coordinates (contained implicitly in ) are constant in the integration. The equation shows that '^ is a function of these constants. 102 THE FUNCTION AND If their values are varied, we shall have by differentiation, if w >2, ^ €=00 ^ F=0 c=ao g f = ao f + da, rg. e'^^^de + da,f^ e'^^de + etc. (323) F=0 F=0 (Since c* vanishes with F, when n > 2, there are no terms due to the variations of the limits.) Hence by (269) -|*A + ^««® = Jd0 + gd«x + gd«, + etc., (324) or, since ^^ ^ = 17, (326) rf^ = l^de — 0^(?ai — ® ^ diotj — etc. (326) Comparing this with (112), we get p; = 5, p- = 5, etc. (327) The first of these equations might be written* \daij€,a \de Ja \daija,q ^ ^ but must not be confounded with the equation which is derived immediately from the identity * See equations (321) and (104). Suffixes are here added to the differential coefficients, to make the meaning perfectly distinct, although the same quan- tities may be written elsewhere without the suffixes, when it is believed that there is no danger of misapprehension. The suffixes indicate the quantities which are constant in the differentiation, the single letter a standing for aU the letters 01,09, etc., or aU except the one which is explicitly raried. THE CANONICAL DISTRIBUTION 108 Moreover, if we eliminate d^^ from (326) by the equation d\ff = %d'^ + ^d® + rf€, (331) obtained by differentiating (325), we get cS = — ®dri — ©-^doi ^%^da^^ etc., (332) or by (321), _d^ = g^+gd^ + g^ + etc. (333) Except for the signs of average, the second member of this equation is the same as that of the identity d4i = ^de + ^dai + ^da^ + etc. (334) For the more precise comparison of these equations, we may suppose that the energy in the last equation is some definite and fairly representative energy in the ensemble. For this purpose we might choose the average energy. It will per- haps be more convenient to choose the most common energy, which we shall denote by 6^. The same suffix will be applied to functions of the energy determined for this value. Our identity then becomes It has been shown that when w > 2. Moreover, since the external coordinates have constant values throughout the ensemble, the values of d^/daj, d^/da^^ etc. vary in the ensemble only on account of the variations of the energy (c), which, as we have seen, may be regarded as sensibly constant throughout the en- semble, when n is very great. In this case, therefore, we may regard the average values dai dot* *' 104 THE FUNCTION AND as practically equivalent to the values relating to the most common energy \dai Jo' \d(h Jo In this case also dE is practically equivalent to de^. We have therefore, for very large values of ti, — rf^ = (?<^ (337) approximately. That is, except for an additive constant, — rj may be regarded as practically equivalent to <^q, when the number of degrees of freedom of the system is very great. It is not meant by this that the variable part of ^+ ^q is numerically of a lower order of magnitude than unity. For when n is very great, — rj and ^q are very great, and we can only conclude that the variable part of v + ^o is insignifi- cant compared with the variable part of rj or of ^q, taken separately. Now we have already noticed a certain correspondence between the quantities and ^ and those which in thermo- dynamics are called temperature and entropy. The property just demonstrated, with those expressed by equation (336), therefore suggests that the quantities <^ and defdij} may also correspond to the thermodynamic notions of entropy and tem- perature. We leave the discussion of this point to a sub- sequent chapter, and only mention it here to justify the somewhat detailed investigation of the relations of these quantities. We may get a clearer view of the limiting form of the relations when the number of degrees of freedom is indefi- nitely increased, if we expand tiie function ^ in a series arranged according to ascending powers of € — €q. This ex- pansion may be written (338) Adding the identical equation THE CANONICAL DISTRIBUTION. 106 we get by (336) n©^ + *~-0- + *> + l^J — 2~" + V &s j "IE" (339) Substituting this value in 6 ® efc, / which expresses the probability that the energy of an unspeci- fied system of the ensemble lies between the limits e' and c", we get a"^^^ JaV5?A--^^Mo-Tr -^ ^*"cfe. (340) f' When the number of degrees of freedom is veiy great, and € — €0 in consequence very small, we may neglect the higher powers and wn^e* ^ ^^ ^ ^ ---+^oj,l^jo-2-^. (341) e « This shows that for a very great nimiber of degrees of fireedom the probability of deviations of energy from the most probable value (e^) approaches the form expressed by the *law of errors.' With this approximate law, we get !5>?+etc. * If a higher degree of accuracy is desired than is afforded hj this formula, it may be multiplied by the series obtained from e by the ordinary formula for the expansion in series of an exponential func- tion. There would be no especial analytical difficulty in taking account of a moderate number of terms of such a series, which would commence 106 THE FUNCTION AND ♦-'o +^0 I ^^ir \* whence ^^T^r.-»--6 _2» - ilog(2»(£-e)*). (344) Now it has been proved in Chapter VII that =- 2 <'« - » We have therefore '(346) approximately. The order of magnitude of ^ — ^^ is there- fore that of log n. This magnitude is mainly constant. The order of magnitude of ^ + ^o ~ i ^**S ** ^ *^* o^ unity. The order of magnitude of ^^ , and therefore of — ^, is that of %.* Equation (388) gives for the first approximation (346) (^-.^)(6-e.) = (-l^* = ^€„ (347) ZrZT« = 5E52 = «^ (348) («^-<^) The members of the last equation have the order of magnitude of ». Equation (338) gives also for the first approximation • Compare (289), (314). THE CANONICAL DISTRIBUTION. 107 whence This is of the order of magnitude of n.* It should be observed that the approximate distribution of the ensemble in energy according to the *law of errors' is not dependent on the particular form of the function of the energy which we have assumed for the index of probability (rj). In any case, we must have ff = ao F=0 6^d€z^lj (361) where e^H^ is necessarily positive. This requires that it shall vanish for e = oo , and also f or e = — oo , if this is a possi- ble value. It has been shown in the last chapter that if e has a (finite) least possible value (which is the usual case) and w > 2, g* will vanish for that least value of e. In general therefore 17 + ^ will have a maximum, which determines the most probable value of the energy. If we denote this value by €0, and distinguish the corresponding values of the func- tions of the energy by the same suf&x, we shall have The probability that an unspecified system of the ensemble * We shall find hereafter that the equation (s-y<-«=-' is exact for any ralue of n greater than 2, and that the equation is exact for any ralue of n greater than 4. 108 THE FUNCTION ^ AND falls within any given limits of energy (c' and e") is repre- sented by f" / e^de. li we expand i; and if> in ascending powers of e — e«, without going beyond tbe squares, the probability that the energy &lls within the given limits takes the form of the ' law of errors ' — e This gives JelU^'Jo-^Kd^Jo^ 2 oe. (363) .o + ^o = ilog[i^(g)^-J,(g)J, (354) ^ ^^=[-(S).-(m]~- (366) We shall have a close approximation in general when the quantities equated in (355) are very small, i. e., when -(S).-(S). is very great. Now when n is very great, — cP/d^ is of the same order of magnitude, and the condition that (356) shall be very great does not restrict very much the nature of the function 17. We may obtain other properties pertaining to average values in a canonical ensemble by the method used for the average of d^jde. Let u be any function of the energy, either alone or with and the external coordinates. The average value of u in the ensemble is determined by the equation r - — ^+^ u=zjue^ de. (367) F=0 THE CANONICAL DISTRIBUTION. 109 Now we have identically jl*-0 + «S7)** de = [«.« J (358) F=0 F=0 Therefore, by the preceding equation K we set w = 1, (a value which need not be excluded,) the second member of this equation vanishes, as shown on page 101, if n > 2, and we get de ~ % = -» (360) as before. It is evident from the same considerations that the second member of (359) will always vanish if n > 2, unless u becomes infinite at one of the limits, in which case a more care- ful examination of the value of the expression will be necessary. To facilitate the discussion of such cases, it will be convenient to introduce a certain limitation in regard to the nature of the system considered. We have necessaxily supposed, in all our treatment of systems canonically distributed, that the system , considered was such as to be capable of the canonical distri- bution with the given value of the modulus. We shall now suppose that the system is such as to be capable of a canonical distribution with any (finite) f modulus. Let us see what cases we exclude by this last limitation. * A more general equation, which is not limited to ensembles canonicallj* distributed, is F«0 where ri denotes, as usual, the index of probability of phase. t The term^ntVe applied to the modulus is intended to exclude the value zero as well as infinity. 110 THE FUNCTION (f> AND The impossibility of a canonical distribution occurs wlien the equation e " F=0 \-9 e ^ de (361) fails to determine a finite value for '^. Evidently the equation cannot make '^ an infinite positive quantity, the impossibility therefore occurs when the equation makes -^ = — oo . Now we get easily from (191) <£ ^ = - ;^ rf©. K the canonical distribution is possible for any values of 0, we can apply this equation so long as the canonical distribu- tion is possible. The equation shows that as O is increased (without becoming infinite) — y^ cannot become infinite unless € simultaneously becomes infinite, and that as is decreased (without becoming zero) — ^lr cannot become infinite unless simultaneously e becomes an infinite negative quantity. The corresponding cases in thermodynamics would be bodies which could absorb or give out an infinite amount of heat without passing certain limits of temperature, when no external work is done in the positive or negative sense. Such infinite values present no analytical difficulties, and do not contradict the general laws of mechanics or of thermodynamics, but they are quite foreign to our ordinary experience of nature. In excluding such cases (which are certainly not entirely devoid of interest) we do not exclude any which are analogous to any actual cases in thermodynamics. We assume then that for any finite value of the second member of (861) has a finite value. When this condition is fulfilled, the second member of (869) will vanish for t^ = «-^ K For, if we set 0' = 20, r=o F=o THE CANONICAL DISTRIBUTION. Ill where '^' denotes the value of '^ for the modulus 0'. Since the last member of this formula vanishes for € = oo , the less value represented by the first member must also vanish for the same value of €. Therefore the second member of (359), which differs only by a constant factor, vanishes at the upper limit. The case of the lower limit remains to be considered. Now The second member of this formula evidently vanishes for the value of €, which gives F= 0, whether this be finite or negative infinity. Therefore, the second member of (359) vanishes at the lower limit also, and we have de ® de or € F=0. (362) This equation, which is subject to no restriction in regard to the value of n, suggests a connection or analogy between the function of the energy of a system which is represented by €~^ V and the notion of temperature in thermodynamics. We shall return to this subject in Chapter XIV. If Wr > 2, the second member of (859) may easily be shown to vanish for any of the following values of u viz. : ^, e^, €, €**, where m denotes any positive number. It will also vanish, when n > 4, for u = d^lde^ and when w > 2A for u = e^ dl^VId^. When the second member of (359) van- ishes, and n > 2, we may write -.fd4> 1\ dff> u du ,«^^. We thus obtain the following equations : Hn >2, <*-«(g-|)=*f-|=-r (^) 112 THE FUNCTION ^ AND 27| = ?f=f. (36.) '(367) Ifn> 4, M^=W-i'=-PT- ♦« Ifn >2A, * di^ = i' d? <370) whence e^ -^^ = ^. (371) Giving h the values 1, 2, 8, etc., we have ^ = L if»>2, ^W-i" "»>*- as akeady obtained. Also d? + ^d?& + V*j -®i ^«>6- (372) * This eqnation may also be obtained from equations (252) and (321). Compare also equation (349) which was deriyed by an approximative method, t Compare equation (850), obtained hy an approximatiye method. THE CANONICAL DISTRIBUTION. 113 If F^ is a continuous increasing function of €^ commencing with F^ = 0, the average value in a canonical ensemble of any function of e^ either alone or with the modulus and the exter* nal coordinates, is given by equation (275), which is identical with (357) except that e, ^, and -^ have the suffix ( )j. The equation may be tiunsf ormed so as to give an equation iden- tical with (359) except for the suffixes. If we add the same suffixes to equation (361), the finite value of its members will determine the possibility of the canonical distribution. From these data, it is easy to derive equations similar to (360), (362)-(372), except that the conditions of their vaUd- ity must be differently stated. The equation 6 'K = requires only the condition already mentioned with respect to V^. This equation corresponds to (362), which is subject to no restriction with respect to the value of n. We may ob- serve, however, that Fwill always satisfy a condition similar to that mentioned with respect to V^.. If Vq satisfies the condition mentioned, and e^« a similar condition, L e., if e^^ is a continuous increasing function of e^, commencing with the value e^« = 0, equations will hold sim- ilar to those given for the case when n > 2, viz., similar to (360), (364)-(368). EspeciaUy important is dffyq 1 deq "" ©' If VqyC^ (oT dVq/d€q^y cPVqldcq^ all satisfy similar conditions, we shall have an equation similar to (369), which was subject to the condition w > 4. And if dPVqjde^ also satisfies a similar condition, we shall have an equation similar to (372), for which the condition was w > 6. Finally, if Vq and h suc- cessive differential coefficients satisfy conditions of the kind mentioned, we shall have equations like (370) and (371) for which the condition was w > 2 A. 8 114 THE FUNCTION ^. These conditions take the place of those given above relat- ing to n. In fact, we might give conditions relating to the differential coefficients of Vj similar to those given relating to the differential coefficients of F^, instead of the conditions relating to n, for the validity of equations (360), (868)-<872). This would somewhat extend the application of the equations. CHAPTER X. ON A DISTRIBUTION IN PHASE CALLED MICROCANONI- CAL IN WHICH ALL THE SYSTEMS HAVE THE SAME ENERGY. Ak important case of statistical equilibrium is that in which all systems of the ensemble have the same energy. We may arrive at the notion of a distribution which will satisfy the necessary conditions by the following process. We may suppose that an ensemble is distributed with a uniform den- sity-in-phase between two Umiting values of the energy, e' and e", and with density zero outside of those limits. Such an ensemble is evidently in statistical equilibrium according to the criterion in Chapter IV, since the densiiy-in-phase may be regarded as a function of the energy. By diminishing the difference of e' and e", we may diminish the differences of energy in the ensemble. The limit of this process gives us a permanent distribution in which the energy is constant. We should arrive at the same result, if we should make the density any function of the energy between the limits e' and e", and zero outside of those limits. Thus, the limiting distri- bution obtamed from the part of a canonical ensemble between two limits of energy, when the difference of the limiting energies is indefinitely diminished, is independent of the modulus, being determined entirely by the energy, and is identical with the limiting distribution obtained from a uniform density between limits of energy approaching the same value. We shall call the limiting distribution at which we arrive by this process microcanonicaL We shall find however, in certain cases, that for certain values of the energy, viz., for those for which e^ is infinite. 116 A PERMANENT DISTRIBUTION IN WHICH this process fails to define a limiting distribution in any such distinct sense as for other values of the energy. The difficulty is not in the process, but in the nature of the case, being entirely analogous to that which we meet when we try to find a canonical distribution in cases when '^ becomes infinite. We have not regarded such cases as affording true examples of the canonical distribution, and we shall not regard the cases in which e^ is infinite as affording true examples of the micro- canonical distribution. We shall in fact find as we go on that in such cases our most important formulae become illusory. The use of formulae relating to a canonical ensemble which contain e^de instead of dp-^ . . . dq^y as in the preceding chapters, amounts to the consideration of the ensemble as divided into an infinity of microcanonical elements. From a certain point of view, the microcanonical distribution may seem more simple than the canonical, and it has perhaps been more studied, and been regarded as more closely related to the fundamental notions of thermodynamics. To this last point we shall return in a subsequent chapter. It is sufficient here to remark that analytically the canonical distribution is much more manageable than the microcanonical We may sometimes avoid difficulties which the microcanon- ical distribution presents by regarding it as the result of the following process, which involves conceptions less simple but more amenable to analytical treatment. We may suppose an e,«MU. distributed ^ . i^t, proportio Jto where «, and e' are constants, and then diminish indefinitely the value of the constant co. Here the density is nowhere zero until we come to the limit, but at the limit it is zero for all energies except e'. We thus avoid the analytical compli- cation of discontinuities in the value of the density, which require the use of integrals with inconvenient limits. In a microcanonical ensemble of systems the energy (e) is constant, but the kinetic energy (e,) and the potential energy ALL SYSTEMS HAVE THE SAME ENERGY. IIT (cq) vary in the different systems, subject of course to the con- dition €p + €q=z€=z constant. (373) Our first inquiries will relate to the division of energy into these two parts, and to the average values of functions of €p and 6g. We shall use the notation i^c to denote an average value in a microcanonical ensemble of energy e. An average value in a canonical ensemble of modulus 0, which has hitherto been denoted by Uj we shall in this chapter denote by tij^, to distinguish more clearly the two kinds of averages. The extension-in-phase within any limits which can be given in terms of €p and e^ may be expressed in the notations of the preceding chapter by the double integral // dVpdVq taken within those limits. If an ensemble of systems is dis- tributed within those limits with a uniform density-in-phase, the average value in the ensemble of any function (u) of the kinetic and potential energies will be expressed by the quotient of integrals // udVpdV, fS dVpdVg Since dVpZ= e^^ de^^ and dep = rfe when e^ is constant, the expression may be written // ue^^dedV^ Sf e^^dedK To get the average value of te in an ensemble distributed microcanonically with the energy 6, we must make the in- tegrations cover the extension-in-phase between the energies 6 and € + de. This gives 118 A PERMANENT DISTRIBUTION IN WHICH _ rg=o But by (299) the value of the integral in the denominator is e^. We have therefore ^='^'^Jue%r,y (374) where «^p and F^ are connected by equation (878), and m, if given as function of Cp, or of €p and e^, becomes in virtue of the same equation a function of €q alone. We shall assume that e^ has a finite value. If w > 1, it is evident from equation (305) that e^ is an increasing function of e, and therefore cannot be infinite for one value of e without being infinite for all greater values of e, which would make — ^|r infinite.* When w > 1, therefore, if we assume that e^ is finite, we only exclude such cases as we found necessary to exclude in the study of the canonical distribution. But when » = 1, cases may occur in which the canonical distribu- tion is perfectly applicable, but in which the formulae for the microcanonical distribution become illusory, for particular val- ues of e, on account of the infinite value of e^. Such failing cases of the microcanonical distribution for particular values of the energy will not prevent us from regarding the canon- ical ensemble as consisting of an infinity of microcanonical ensembles, f * See equation (322). t An example of the failing ease of the microcanonical distribution is afforded hj a material point, under the influence of gravity, and constrained to remain in a vertical circle. The failing case occurs when the energy is just sufficient to carry the material point to the highest point of the circle. It will be observed that the difficulty is inherent in the nature of the case, and is quite independent of the mathematical formulae. The nature of the difficulty is at once apparent if we try to distribute a finite number of ALL SYSTEMS HAVE THE SAME ENERGY. 119 From the last equation, with (298), we get e-*'r. = ejVpdVq = e r. (376) ri=o But by equations (288) and (289) e *''T^ = ^6,. (376) Therefore e V= a"*' FjJ. = ? ^ . (377) Again, with the aid of equation (801), we get Sl-'/t'*'^"'"^' '■'"> FgCzO ^ if n > 2. Therefore, by (289), de These results are interesting on account of the relations of the functions e"^ V and -^ to the notion of temperature in thermodynamics, — a subject to which we shall return here- after. They are particular cases of a general relation easily deduced from equations (806), (874), (288) and (289). We have d^r d^ ^ v..„ The equation may be written -4>d^V -4>Y -^pd^V„ p e rf€* = .-*/,-*^.^.r.. F,=0 material points with this particular value of the energy as nearly as possible in statistical equilibrium, or if we ask : What is the probability that a point taken at random from an ensemble in statistical equilibrium with this yalue of the energy will be found in any specified part of the circle? 120 A PERMANENT DISTRIBUTION IN WHICH We have therefore -^dJ*r -0/>d*K, 6 r(+n) if A < J w + !• For example, when n is even, we may make A = J w, which gives, with (307), fi .2 ^-i Since any canonical ensemble of systems may be regarded as composed of microcanonical ensembles, if any quantities u and V have the same average values in every microcanonical ensemble, they will have the same values in every canonical ensemble. To bring equation (880) formally under this rule, we may observe that the first member being a function of € is a constant value in a microcanonical ensemble, and therefore identical with its average value. We get thus the general equation 6 d^ de^" ifA 2. The last two equations give for a canonical ensemble, if w > 2, (l-|)^eV^e = l- (386) The corresponding equations for a microcanonical ensemble give, if » > 2, 2\ — , — „ d4> (i-^)^.v^.= rf log V (386) * See eqnation (202). ALL SYSTEMS HAVE THE SAME ENERGY. 121 which shows that d dlog V approaches the value unity when n is very great. If a system consists of two -parts, having separate energies, we may obtain equations similar in form to the preceding, which relate to the system as thus divided.* We shall distinguish quantities relating to the parts by letters with suffixes, the same letters without suffixes relating to the whole system. The extension-in-phase of the whole system within any given limits of the energies may be represented by the double integral // dVxdVi taken within those limits, as appears at once from the defini- tions of Chapter VIII. In an ensemble distributed with uniform density within those limits, and zero density outside, the average value of any function of €i and €« is given by the quotient // udVidVi IS' dVidV^ which may also be written f IS"" sp- dedVa If we make the limits of integration € and e + rfe, we get the * If this condition is rigorously fulfilled, the parts will haye no influence on each other, and the ensemble formed by distributing the whole micro- canonically is too arbitrary a conception to have a real interest The prin- cipal interest of. the equations which we shall obtain will be in cases in which the condition is approximately fulfilled. But for the purposes of a theoretical discussion, it is of course convenient to make such a condition absolute. Compare Chapter lY, pp. 35 ft., where a similar condition is con- sidered in connection with canonical ensembles. t Where the analytical transformations are identical in form with those on the preceding pages, it does not appear necessary to giye aU the steps with the same detail. 120 A PERMANENT DISTRIBUTION IN WHICH We have therefore -^d*r -0/>rf*K. d^ = e rf€p* T(in) t ' r(i7i-A + i) i7=%, (380) if A < J w + 1. For example, when n is even, we may make A = ^ n, which gives, with (307), fi 5 ^-i Since any canonical ensemble of systems may be regarded as composed of microcanonical ensembles, if any quantities u and V have the same average values in every microcanonical ensemble, they will have the same values in every canonical ensemble. To bring equation (380) formally under this rule, we may observe that the first member being a function of 6 is a constant value in a microcanonical ensemble, and therefore identical with its average value. We get thus the general equation e wr dt^ = e rfe/ i= (382) ifA 2. The last two equations give for a canonical ensemble, if w > 2, (i - 1) ^e v=n« = 1- (386) The corresponding equations for a microcanonical ensemble give, if » > 2, 1 - -) ^. vH. = ^13^, (386) * See equation (202). ALL SYSTEMS HAVE THE SAME ENERGY. 121 which shows that d dlog V approaches the value unity when n is very great. If a system consists of two -parts, having separate energies, we may obtain equations similar in form to the preceding, which relate to tiie system as thus divided.* We shall distinguish quantities relating to the parts by letters with suffixes, the same letters without suffixes relating to the whole system. The extension-in-phase of the whole system within any given limits of the energies may be represented by the double integral dVxdVi ff- taken within those limits, as appears at once from the defini- tions of Chapter VIII. In an ensemble distributed with uniform density within those limits, and zero density outside, the average value of any function of Ci and €« is given by the quotient f fudVidVa fJdV^dV^ which may also be written f SI"' C fe^dedVt If we make the limits of integration e and e + de, we get the * If this condition is rigorously fulfilled, the parts will hare no influence on each other, and the ensemble formed by distributing the whole micro- canonically is too arbitrary a conception to have a real Interest The prin- cipal interest of. the equations which we shaU obtain will be in cases in which the condition is approximately fulfilled. But for the purposes of a theoretical discussion, it is of course convenient to make such a condition absolute. Compare Chapter lY, pp. 36 ff., where a similar condition is con- sidered in connection with canonical ensembles. t Where the analytical transformations are identical in form with those on the preceding pages, it does not appear necessary to give aU the steps with the same detail. 122 A PERMANENT DISTRIBUTION IN WHICH average value of u in an ensemble in which the whole system is microcanonically distributed in phase, viz., (387) Fi=o where ^^ ^^^ ^ ^^^ connected by the equation 61 + €3 = constant = 6, (388) and u, if given as function of 61, or of €1 and €,, becomes in virtue of the same equation a function of €% alone.* Thus 7^ = e"^JrirfF„ (389) 6 "P^ r,H=o This requires a similar relation for canonical averages (390) Again But if n^ > 2, 6^1 vanishes for V^ = 0,t and f2=€ f2=e ^ r,=o r,=o Hence, if t^^ > 2, and Wa > 2, (391) (392) (393) c?<^ di de d€i 3^ € dCi (394) * In the applications of the equation (387), we cannot obtain all the results corresponding to those which we have obtained from equation (374), because 4>p is a known function of €p, while i must be treated as an arbitrary func- tion of ti, or nearly so. t See Chapter VHI, equations (305) and (316). ALL SYSTEMS HAVE THE SAME ENERGY. 12a J. cUf} d(f>i cUfi^ and ^— ^ ® de dci '2 d€> 2 (395) We have compared certain functions of the energy of the whole system with average values of similar functions of the kinetic energy of the whole system, and with average values of similar functions of the whole energy of a part of the system. We may also compare the same functions with average values of the kinetic energy of a part of the system. We shall express the total, kinetic, and potential energies of the whole system by e, €p, and e^, and the kinetic energies of the parts by e-^ and e^j,. These kinetic energies are necessarily sep- arate : we need not make any supposition concerning potential energies. The extension-in-phase within any limits which can be expressed in terms of €^ Cj^, Cgp may be represented in the notations of Chapter VIII by the triple integral JJJdVr^dV,,dV^ taken within those limits. And if an ensemble of systems is distributed with a uniform density within those limits, the average value of any function of e^ ej^, c^ will be expressed by the quotient J J JudVipdKpdV^ dVipdVipdV^ or Iff JJJe^^dedV^pdV, To get the average value of u for a microcanonical distribu- tion, we must make the limits e and e + de. The denominator in this case becomes e^ de^ and we have €a — € *2P~** — '* lZ], = e*J J ue*"" dV^dV^, (396) rj=o €s»,=o 124 A PERMANENT DISTRIBUTION IN WHICH where ^^ F^, and V^ are connected by the equation €ip + €,p + €« = constant = e. Accordingly 6 r^ FipL = e-^J fr^,dv,,dv, = e-^r, (397) and we may write -4> j^ -^ip T^ 6 r= 6 Vip ni ^^. (398) and = 7^. =7*^« = r^e = |-ii;W = ^^e. (399) 71-1 7«2 Again, if w^ > 2, f,=f fjp=f-fg dip de ip F«=0 f2p=0 F,=0 Hence, if w^ > 2, and n^ > 2, a

i wV'2p d^ 2p = (ini-l)v=^ = (in,~l)i;7=^„ (401) 1 o^ dip (;^e e C?€ ip — ?s e c^e 2p = (*nx~l)i7=T^=(in,-l)6l7^^. (402) We cannot apply the methods employed in the preceding pages to the microcanonical averages of the (generalized) forces A^j -4^, etc., exerted by a system on external bodies, since these quantities are not functions of the energies, either kinetic or potential, of the whole or any part of the system. We may however use the method described on page 116. ALL SYSTEMS HAVE THE SAME ENERGY. 125 Let us imagine an ensemble of systems distributed in phase according to the index of probability where ^ is any constant which is a possible value of the energy, except only the least value which is consistent with the values of the external coordinates, and c and o) are other constants. We have therefore "Je *^ dp^...dq^ = l, (403) nhiifMi . . . j e ^ dpi . .. dq^ (404) phMM or again e = C e "^ de. (405) F=0 From (404) we have c:?6' = /,../ 2 a~-^i^ dpi...dq^ ddi phases =j 2^^.e -* 2 on account of the factor e^.) We have therefore from these equations F=0 F=0 or F=0 Si^-^t-^)'"'^--''- w That is : the average value in the ensemble of the quantity represented by the principal parenthesis is zero. This must ALL SYSTEMS HAVE THE SAME ENERGY. 127 be trae for any value of ©. If we diminish «, the average value of the parenthesis at the limit when o) vanishes becomes identical with the value for 6 = 6'. But this may be any value of the energy, except the least possible. We have therefore unless it be for the least value of the energy consistent with the external coordinates, or for particular values of the ex- ternal coordinates. But the value of any term of this equar tion as determined for particular values of the energy and of the external co(3rdinates is not distinguishable from its value as determined for values of the energy and external coordinates indefinitely near those particular values. The equation therefore holds without limitation. Multiplying by e^, we get «*^+^.e*^ = e*^=^=/^. (411) a€ ' ae dai dai daide ^ ' The integral of this equation is 3n.e* = g'+i?'x, (412) where jP^ is a function of the external coordinates. We have an equation of this form for each of the external coordinates. This gives, with (266), for the complete value of the differen- tial of V rfr=e^ifc + (/3^«-i^i)(fai + (6^3;;ic-i^2)da, + etc., (413) or rfF"= ^ {de. + T^dax + 3^rfaa + etc.) —-F^doy — F^da^ — etc. (414) To determine the values of the functions jF\ , jRj » etc., let us suppose ^11,^2, etc. to vary arbitrarily, while e varies so as always to have the least value consistent with the values of the external coordinates. This will make ]^= 0, and dV'=- 0. If n < 2, we shall have also e^ = 0, which wiU give J^lrrO, i^a = 0, etc. (416) 128 THE MICROCANONICAL DISTRIBUTION. The result is the same for any value of n. For in the varia- tions considered the kinetic energy will be constantly zero, and the potential energy will have the least value consistent with the external coordinates. The condition of the least possible potential energy may limit the ensemble at each in- stant to a single configuration, or it may not do so ; but in any case the values of A^ , J^, etc. will be the same at each instant for all the systems of the ensemble,* and the equation de + Aiddi -f A^da^ -f etc. = will hold for the variations considered. Hence the functions -^1 » -^2 » ®*^* v^^i^l^ i^ ^^y c^®> ^^^ ^^ have the equation d V= e^de + e^Z^rfoi + ^IQ^diH + etc., (416) de +^iL<^ + -271 c/a« + etc. or rf log F = _^^ , (417) or again de = e^ Vd\Q% V —l^doi — l^^da^ — etc. (418) It will be observed that the two last equations have the form of the fundamental differential equations of thermodynamics, e-^'V corresponding to temperature and log V to entropy. We have already observed properties of e*"^]^ suggestive of an analogy with temperature, f The significance of these facts will be discussed in another chapter. The two last equations might be written more simply ,_. de + 37[f ^J. . .Jfje^'dpi ...dq,, (419) phMes phases where 1; is a function of the energy, and A17 a function of the phase, which are subject to the conditions that aU aU j. . . fe'^^^'dpi ...dq^=zj... Je''dpi...dq^ = 1, (420) phases phases and that for any value of the energy (/) f. . . fe'^^'^dpi ...dq^^C... Ce^'dpi . . . dq^. (421) Equation (420) expresses the general relations which 17 and V + ^v must satisfy in order to be indices of any distributions, and (421) expresses the condition that they give the same distribution in eneigy. 9 130 MAXIMUM AND MINIMUM PROPERTIES. Since 17 is a function of the energy, and may therefore be re- garded as a constant within the limits of integration of (421), we may multiply by 97 under the integral sign in both mem- bers, which gives I . • . I i;e dpi . . . dq^ = j , . , j rjC dpi . . . dq^. Since this is true within the limits indicated, and for every value of e', it will be true if the integrals are taken for all phases. We may therefore cancel the corresponding parts of (419), which gives J. . .(^-qe^^'^dp^ . . . <%n > 0. (422) phases But by (420) this is equivalent to aU f. . . TcAiye^" -*- 1 - e^'^e^dpi ...dq^>0. (423) phases Now A77 e^'' + 1 — e^"^ is a decreasing function of A17 for nega- tive values of A17, and an increasing function of A17 for positive values of A17. It vanishes for A17 = 0. The expression is therefore incapable of a negative value, and can have the value only for Arj = 0. The inequality (423) will hold therefore unless A17 = for all phases. The theorem is therefore proved. Theorem IL If an ensemble of systems is canonically dis- tributed in phase, the average index of probability is less than in any other distribution of the ensemble having the same average energy. For the canonical distribution let the index be ("^ — c)/®» and for another having the same average energy let the index be (-^ — e)/0 + A77, where A77 is an arbitrary function of the phase subject only to the limitation involved in the notion of the index, that MAXIMUM AND MINIMUM PROPERTIES. 131 aU ^c aU ^— c /.../« ® dpi. . .dq^^ j . . . J e ^ dpi. . . dq^=:l, phases phases (424) and to that relating to the constant average energy, that an ^c all ^e /...lee® dpi. . .dq^ = j . . . j ee ^ dpx . . . dq^. (425) phases phases It is to be proved that aU ^6 /• • •/(! - 1 + ^'?)«'*"^ ''^i'x . . . rf?. > phases an ^_f phases Now in virtue of the first condition (424) we may cancel the constant term -^ /© in the parentheses in (426), and in virtue of the second condition (425) we may cancel the term 6/0. The proposition to be proved is thus reduced to aU ^f—f /• • •/' +Ai| ^rje ^ dpi . . . dq^^ > 0, phases which may be written, in virtue of the condition (424), aU ^-c r. ..n^rje''^ + 1 - e^'^)e^dpi ...dq^>0. (427) phases In this form its truth is evident for the same reasons which appUed to (423). Theorem HI. If is any positive constant, the average value in an ensemble of the expression i; + ^ / ® C'? denoting as usual the index of probability and e the energy) is less when the ensemble is distributed canonically with modulus 0, than for any other distribution whatever. In accordance with our usual notation let us write (-^ — e) /0 for the index of the canonical distribution. In any other distribution let the index be (-^ — c)/0 + A17. 182 MAXIMUM AND MINIMUM PROPERTIES. In the canonical ensemble rj + e/S has the constant value yjr/S; in the other ensemble it has the value yft/S -\- Arj. The proposition to be proved may therefore be written • • •Vm relate to one part of the system, and j.^^ > • • • Jn? i>m+i » • • ^Pn to the other. If the index of probability for the whole system is denoted by 17, the probability that the phase of an unspecified system lies within any given limits is expressed by the integral r. . . l^d^x . . . ^fi'n (431) taken for those limits. If we set / . . .J e'cip^i . . . djp^dq^^x . . . (^, = 6% (432) where the integrations cover all phases of the second system, and J. .. je'* dpi...dp^dqi... c?2'« = «% (433) 134 MAXIMUM AND MINIMUM PROPERTIES. where the integrations cover all phases of the first qrstem, the integral (481) will reduce to the form J. . .je^ dpi.. . dp^dqi ...dq^ (434) when the limits can be expressed in terms of the coordinates and momenta of the first part of the system. The same integral will reduce to / . . . / e'^ dp^^ .. .dp^ dq^i . . . dq^ (435) when the limits can be expressed in terms of the co()*rdinates and momenta of the second part of the system. It is evident that Tji and rj^ are the indices of probability for the two parts of the system taken separately. The main proposition to be proved may be written j . . . I ViC^ dpi. . . dq^ +/.../ rj^e^ dp^i . . .dq^< I . . . / 17 e' dpi . . . dq^y (436) where the first integral is to be taken over all phases of the first part of the system, the second integral over all phases of the second part of the system, and the last integral over all phases of the whole system. Now we have r. . . fe'dpi ...dq^ = ly (437) r. . . fe^dpi ...dq^=zl, (438) and / • • • / ^'^^i^'M-i . . . ^S'n = 1, (439) where the limits cover in each case all the phases to which the variables relate. The two last equations, which are in them- selves evident, may be derived by partial integration from the first. MAXIMUM AND MINIMUM PROPERTIES. 135 It appears from the definitions of i/j and v^ ^^^^t (436) may also be written j . . . j Tje'^ dpi... dq^, (440) or I . • • / (i? — ^1 — V^^'^^Pi • • • <^» ^ 0, where the integrations cover all phases. Adding the equation r. . .fe'^^dpi . . . cJ?, = l, (441) which we get by multiplying (438) and (439), and subtract- ing (437), we have for the proposition to be proved /. . 'f[(ri-Vi-V2)e'' + 6'^+'^ - e'] rfi>i ...dq,>0. (442) phaaes Let w = ^ — 171 — Vr- (443) The main proposition to be proved may be written aU r. . . r(w^ + 1 - e'^e'^-^dpi ...dq^>0. (444) phases This is evidently true since the quantity in the parenthesis is incapable of a negative value.* Moreover the sign = can hold only when the quantity in the parenthesis vanishes for all phases, i. e.y when t£ = for all phases. This makes V = vi + Va iov ail phases, which is the analytical condition which expresses that the distributions in phase of the two parts of the system are independent. Theorem VIIL If two or more ensembles of systems which are identical in nature, but may be distributed differently in phase, are united to form a single ensemble, so that the prob- ability-coefficient of the resulting ensemble is a linear function * See Theorem I, where this is prored of a similar expression. 186 MAXIMUM AND MINIMUM PROPERTIES. of the probability-coefficients of the original ensembles, the average index of probability of the resulting ensemble cannot be greater than the same linear function of the average indices of the original ensembles. It can be equal to it only when the original ensembles are similarly distributed in phase. Let PiyP^t etc. be the probability-coefficients of the original ensembles, and P that of the ensemble formed by combining them ; and let N^^N^^ etc. be the numbers of systems in the original ensembles. It is evident that we shall have P = CiPi -f CaP, -f etc. = 5 (ciPi), (445) N N where ^"si^' ^~Sl^' ®*^' ^^^^ The main proposition to be proved is that all all phaaea L. phaaea -i (447) ■a or r... r[2(oiP,logPi)-PlogP]0. (448) phaaea If we set Gi = Pi log Pi - Pi log P - Pi + P Q^ will be positive, except when it vanishes for P^ = P. To prove this, we may regard P^ and P as any positive quantities. Then VdPiVp~Pi' Since Q^ and dQJdPi vanish for Pj = P, and the second differential coefficient is always positive, Qi must be positive except when Pj = P. Therefore, if Q^, etc. have similar definitions, S (ci Qi) > 0. (449) MAXIMUM AND MINIMUM PROPERTIES. 137 But since 2 (ci Pi) = P and 2 Ci = 1, S(ci Qi) = S(cJi Pi log Pi) ~Plog P. (460) This proves (448), and shows that the sign = will hold only when Pi = P, Fi = Pf etc. for all phases, f . e., only when the distribution in phase of the original ensembles are all identical. Theorem ZX A uniform distribution of a given number of systems within given limits of phase gives a less average index of probability of phase than any other distribution. heir) be the constant index of the uniform distribution, and rj + Atj the index of some other distribution. Since the num- ber of systems within the given limits is the same in the two distributions we have r. . . Ce"^^"^ dpi...dq^=zC,.. Ce^dpi . . . dq^, (451) where the integrations, like those which follow, are to be taken within the given limits. The proposition to be proved may be written r. ..J(ri + ^v) ^"^^"^ dpi...dq^> C... fve^dpi...dq^, (452) or, since rj is constant, / • • • / (^7 + ^v) ^ ^^Pi • • • ^In >/•••/ 1? ^P\ • • • ^n- (453) In (451) also we may cancel the constant factor e\ and multiply by the constant fa.ctor (i; + 1). This gives / • • • / (^ + 1) « ^ dpi . . . dqn = I . . . j (rj + 1) dpi . . . dq^. The subtraction of this equation will not alter the inequality to be proved, which may therefore be written I ... I (Aiy — 1) 6 '^ (ipi . . • dq^^ >/•••/" ^Pi • • • <^n 188 MAXIMUM AND MINIMUM PROPERTIES. or r. ..C(£iri6^'' •-e^'' + l)dp^...dq^> 0. (464) Since the parenthesis in this expression represents a positive Yalue, except when it vanishes for Arf = 0, the integral will be positive unless A, vanishes everywhere within the limita, which would make the difference of the two distributions vanish. The theorem is therefore proved. CHAPTER Xn. ON THE MOTION OF SYSTEMS AND ENSEMBLES OF SYS- TEMS THROUGH LONG PERIODS OP TIME. An important question which suggests itself in regaid to any case of dynanll motion is wh^er the system consider^ wiU return in the course of time to its initial phase, or, if it will not return exactly to that phase, whether it will do so to any required degree of approximation in the course of a suffi- ciently long time. To be able to give even a partial answer to such questions, we must know something in regard to the dynamical nature of the system. In the foUowing theorem, the only assumption in this respect is such as we have found necessary for the existence of the canonical distribution. If we imagine an ensemble of identical systems to be distributed with a uniform density throughout any finite extension-in-phase, the number of the systems which leave the extension-in-phase and will not return to it in the course of time is less than any assignable fraction of the whole number; provided^ that the total extension-in-phase for the systems considered between two limiting values of the energy is finite, these limiting values being less and greater respec- tively than any of the energies of the first-mentioned exten- sion-in-phase. To prove this, we observe that at the moment which we call initial the systems occupy the given extension-in-phase. It is evident that some systems must leave the extension immediately, unless all remain in it forever. Those systems which leave the extension at the first instant, we shall call tiie front of the ensemble. It will be convenient to speak of this front as generating the extension-in-phase through which it passes in the course of time, as in geometry a surface is said to 140 MOTION OF SYSTEMS AND ENSEMBLES generate the volume through which it passes. In equal times tiie front generates equal extensions in phase. This is ai> immediate consequence of the principle of conservation oj extemion-irirpJiasej unless indeed we prefer to consider it as a slight variation in the expression of that principle. For in two equal short intervals of time let the extensions generated be A and B. (We make the intervals short simply to avoid the complications in the enunciation or interpretation of the principle which would arise when the same extension-in-phase is generated more than once in the interval considered.) Now if we imagine that at a given instant systems are distributed throughout the extension J., it is evident that the same systems will after a certain time occupy the extension J?, which is therefore equal to -4 in virtue of the principle cited. The front of the ensemble, therefore, goes on generating equal extensions in equal times. But these extensions are included in a fmite extension, viz., that bounded by certain limiting values of the energy. Sooner or later, therefore, the front must generate phases which it has before generated. Such second generation of the same phases must commence with the initial phases. Therefore a portion at least of the front must return to the original extension-in-phase. The same is of course true of the portion of the ensemble which follows that portion of the front through the same phases at a later time. It remains to consider how large the portion of the ensemble is, which will return to the original extension-in-phase. There can be no portion of the given extension-in-phase, the systems of which leave the extension and do not return. For we can prove for any portion of the extension as for the whole, that at least a portion of the systems leaving it will return. We may divide the given extension-in-phase into parts as follows. There may be parts such that the systems within them will never pass out of them. These parts may indeed constitute the whole of the given extension. But if tiie given extension is very small, these parts will in general be non- existent. There may be parts such that systems within them THROUGH LONG PERIODS OF TIME. 141 will all pass out of the given extension and all return within it. The whole of the given extension-in-phase is made up of parts of these two kinds. This does not exclude the possi- bility of phases on the boundaries of such parts, such that systems starting with those phases would leave the extension and never return. But in the supposed distribution of an ensemble of systems with a uniform density-in-phase, such systems would not constitute any assignable fraction of the whole number. These distinctions may be illustrated by a very simple example. K we consider the motion of a rigid body of which one point is fixed, and which is subject to no forces, we find three cases. (1) The motion is periodic. (2) The system will never return to its original phase, but will return infinitely near to it. (3) The system will never return either exactly or approximately to its original phase. But if we consider any extension-in-phase, however small, a system leaving that extension will return to it except in the case called by Poinsot * singular,' viz., when the motion is a rotation about an axis lying in one of two planes having a fixed position relative to the rigid body. But all such phases do not constitute any true exten^io'nArwphaBe in the sense in which we have defined and used the term.* In the same way it may be proved that the systems in a canonical ensemble which at a given instant are contained within any finite extension-in-phase will in general return to * An ensemble of Bystems distributed in phase is a less simple and ele- mentary conception than a single system. But by the consideration of suitable ensembles instead of single systems, we may get rid of the incon- yenience of having to consider exceptions formed by particular cases of the integral equations of motion, these cases simply disappearing when the ensemble is substituted for the single system as a subject of study. This is especially true when the ensemble is distributed, as in the case called canonical, throughout an extension-in-phase. In a less degree it is true of the microcanonical ensemble, which does not occupy any extension-in-phase, (in the sense in which we have used the term,) although it is convenient to regard it as a limiting case with respecf to ensembles which do, as we thus gain for the subject some part of the analytical simplicity which belongs to the theory of ensembles which occupy true extensions-in-phase. 142 MOTION OF SYSTEMS AND ENSEMBLES that extension-in-phase, if they leave it, the exceptions, i. €., the number which pass out of the extension-in-phase and do not return to it, being less than any assignable fraction of the whole number. In other words, the probability that a system taken at random from the part of a canonical ensemble which is contained within any given extension-in-phase, will pass out of that extension and not return to it, is zero. A similar theorem may be enunciated with respect to a microcanonical ensemble. Let us consider the fractional part of such an ensemble which lies within any given limits of phase. This fraction we shall denote by F. It is evidently constant in time since the ensemble is in statistical equi- librium. The systems within the limits will not in general remain the same, but some wiU pass out in each unit of time while an equal number come in. Some may pass out never to return within the limits. But the number which in any time however long pass out of the limits never to return will not bear any finite ratio to the number within the limits at a given instant. For, if it were otherwise, let / denote the fraction representing such ratio for the time T. Then, in the time T^ the number which pass out never to return will bear the ratio fF to the whole number in the ensemble, and in a time exceeding T/(^fF) the number which pass out of the limits never to return would exceed the total nimiber of systems in the ensemble. The proposition is therefore proved. This proof will apply to the cases before considered, and may be regarded as more simple than that which was given. It may also be applied to any true case of statistical equilib- rium. By a true case of statistical equilibrium is meant such as may be described by giving the general value of the prob- ability that an unspecified system of the ensemble is con- tained within any given limits of phase.* * An ensemble in which the systems are material points constrained to move in vertical circles, with jnst enough energy to carry them to the highest points, cannot afford a true example of statistical equilibrium. For any other yalue of the energy than the critical yalue mentioned, we might THROUGH LONG PERIODS OF TIME. 143 Let us next consider whether an ensemble of isolated systems has any tendency in the course of time toward a state of statistical equilibrium. There are certain functions of phase which are constant in time. The distribution of the ensemble with respect to the values of these functions is necessarily invariable, that is, the number of systems within any limits which can be specified in terms of these functions cannot vary in the course of time. The distribution in phase which without violating this condition gives the least value of the average index of probability of phase (^) is unique, and is that in which the in yarious ways describe an ensemble in statistical equilibrium, while the same language applied to the critical value of the energy would fail to do so. Thus, if we should say that the ensemble is so distributed that the probability that a system is in any given part of the circle is proportioned to the time which a single system spends in that part, motion in either direc- tion being equally probable, we should perfectly define a distribution in sta- tistical equilibrium for any value of the energy except the critical value mentioned above, but for this value of the energy all the probabilities in question would vanish unless the highest point is included in the part of the circle considered, in which case the probability is unity, or forms one of its limits, in which case the probability is indeterminate. Compare the foot-note on page 118. A stiU more simple example is afforded by the uniform motion of a material point in a straight line. Here the impossibility of statistical equi- librium is not limited to any particular energy, and the canonical distribu- tion as well as the microcanonical is impossible. These examples are mentioned here in order to show the necessity of caution in the application of the above principle, with respect to the question whether we have to do with a true case of statistical equilibrium. Another point in respect to which caution must be exercised is that the part of an ensemble of which the theorem of the return of systems is asserted should be entirely defined by limits within which it is contained, and not by any such condition as that a certain function of phase shall have a given value. This is necessary in order that the part of the ensemble which is considered should be any assignable fraction of the whole. Thus, if we have a canonical ensemble consisting of material points in vertical circles, the theorem of the return of systems may be applied to a part of the ensemble defined as contained in a given part of the circle. But it may not be applied in all cases to a part of the ensemble defined as contained in a given part of the circle and having a g^ven energy. It would, in fact, express the exact opposite of the truth when the given energy is the critical value mentioned above. 144 MOTION OF SYSTEMS AND ENSEMBLES index of probability (i;) is a f imctidn of the functions men- tioned.* It is therefore a permanent distribution,f and the only permanent distribution consistent with the invariability of the distribution with respect to the functions of phase which are constant in time. It would seem, therefore, that we might find a sort of meas- ure of the deviation of an ensemble from statistical equilibrium in the excess of the average index above the minimum which is consistent with the condition of the invariability of the distri- bution with respect to the constant functions of phase. But we have seen that the index of probability is constant in time for each system of the ensemble. The average index is there- fore constant, and we find by this method no approach toward statistical equilibrium in the course of time. Yet we must here exercise great caution. One* function may approach indefinitely near to another function, while some quantity determined by the first does not approach the corresponding quantity determined by the second. A line joining two points may approach indefinitely near to the straight line joining them, while its length remains constant. We may find a closer analogy with the case under considera- tion in the effect of stirring an incompressible liquid.^ In space of 2 71 dimensions the case might be made analyti- cally identical with that of an ensemble of systems of n degrees of freedom, but the analogy is perfect in ordinary space. Let us suppose the liquid to contain a certain amount of coloring matter which does not affect its hydrodynamic properties. Now the state in which the density of the coloring matter is uniform, L e., the statt, of perfect mixture, which is a sort of state of equilibrium in this respect that the distribu- tion of the coloring matter in space is not affected by the internal motions of the liquid, is characterized by a minimum « See Chapter XI, Theorem IV. t See Chapter IV, sub init, X By liquid is here meant the continuous body of theoretical hydrody- namics, and not anything of the molecular structure and molecular motions of real liquids. THROUGH LONG PERIODS OF TIME. 145 value of the average square of the density of the coloring matter. Let us suppose, however, that the coloring matter is distributed with a variable density. K we give the liquid any motion whatever, subject only to the hydrodynamic law of incompressibility, — it may be a steady flux, or it may vary with the time, — the density of the coloring matter at any same point of the liquid wiU. be unchanged, and the average square of this density wiU. therefore be unchanged. Yet no fact is more familiar to us than that stirring tends to bring a liquid to a state of uniform mixture, or uniform densities of its components, which is characterized by minimum values of the average squares of these densities. It is quite true that in the physical experiment the result is hastened by the process of diffusion, but the result is evidently not dependent on that process. The contradiction is to be traced to the notion of the dermty of the coloring matter, and the process by which this quantity is evaluated. This quantity is the limiting ratio of the quantity of the coloring matter in an element of space to the volume of that element. Now if we should take for our ele- ments of volume, after any amount of stirring, the spaces occupied by the same portions of the liquid which originally occupied any given system of elements of volimie, the densi- ties of the coloring matter, thus estimated, would be identical with the original densities as determined by the given system of elements of volume. Moreover, if at the end of any finite amount of stirring we should take our elements of volume in any ordinary form but suflficiently small, the average square of the density of the coloring matter, as determined by such element of volume, would approximate to any required degree to its value before the stirring. But if we take any element of space of fixed position and dimensions, we may continue the stirring so long that the densities of the colored liquid estimated for these fixed elements will approach a uniform limit, viz., that of perfect mixture. The case is evidently one of those in which the limit of a limit has different values, according to the order in which wo 10 146 MOTION OF SYSTEMS AND ENSEMBLES apply the processes of taking a limit. If treating the elements of volume as constant, we continue the stirring indefinitely, we get a uniform density, a result not affected by making the elements as small as we choose ; but if treating the amount of stirring as finite, we diminish indefinitely the elements of volume, we get exactly the same distribution in density as before the stirring, a result which is not affected by con- tinuing the stirring as long as we choose. The question is largely one of language and definition. One may perhaps be allowed to say that a finite amount of stirring will not affect the mean square of the density of the coloring matter, but an infinite amount of stirring may be regarded as producing a condition in which the mean square of the density has its minimimi value, and the density is uniform. We may cer- tainly say that a sensibly uniform density of the colored com- poi\ent may be produced by stirring. Whether the time required for this result would be long or short depends upon the nature of the motion given to the liquid, and the fineness of our method of evaluating the density. All this may appear more distinctly if we consider a special case of liquid motion. Let us imagine a cylindrical mass of liquid of which one sector of 90° is black and the rest white. Let it have a motion of rotation about the axis of the cylinder in which the angular velocity is a function of the distance from the axis. In the course of time the black and the white parts would become drawn out into thin ribbons, which would be wound spirally about the axis. The thickness of these rib- bons would diminish without limit, and the liquid would there- fore tend toward a state of perfect mixture of the black and white portions. That is, in any given element of space, the proportion of the black and white would approach 1 : 3 as a limit. Yet after any finite time, the total volume would be divided into two parts, one of which would consist of the white liquid exclusively, and the other of the black exclusively. If the coloring matter, instead of being distributed initially with a uniform density throughout a section of the cylinder, were distributed with a density represented by any arbitrary f unc- THROUGH LONG PERIODS OF TIME, 147 tion of the cylindrical coordinates r, 6 and z, the effect of the same motion continued indefinitely would be an approach to a condition in which the density is a function of r and z alone. In this limiting condition, the average square of the density would be less than in the original condition, when the density was supposed to vary with ^, although after any finite time the average square of the density would be the same as at first. If we limit our attention to the motion in a single plane perpendicular to the axis of the cylinder, we have something which is almost identical with a diagrammatic representation of the changes in distribution in phase of an ensemble of systems of one degree of freedom, in which the motion is periodic, the period varying with the energy, as in the case of a pendulum swinging in a circular arc. If the coordinates and momenta of the systems are represented by rectangu- lar coordinates in the diagram, the points in the diagram representing the changing phases of moving systems, will move about the origin in closed curves of constant energy. The motion will be such that areas bounded by points repre- senting moving systems will be preserved. The only differ- ence between the motion of the liquid and the motion in the diagram is that in one case the paths are circular, and in the other they differ more or less from that form. When the energy is proportional to jt?^ + g^ the curves of constant energy are circles, and the period is independent of the energy. There is then no tendency toward a state of sta- tistical equilibrium. The diagram turns about the origin with- out change of form. This corresponds to the case of liquid motion, when the liquid revolves with a uniform angular velocity like a rigid solid. The analogy between the motion of an ensemble of systems in an extension-in-phase and a steady current in an incompres- sible liquid, and the diagrammatic representation of the case of one degree of freedom, which appeals to our geometrical in- tuitions, may be suiE&cient to show how the conservation of density in phase, which involves the conservation of the 148 MOTION OF SYSTEMS AND ENSEMBLES average value of the index of probability of phase, is ccoisist- ent with an approach to a limiting condition in which that average value is less. We might perhaps fairly infer from such considerations as have been adduced that an approach to a limiting condition of statistical equilibrium is the general rule, when the initial condition is not of that character. But the subject is of such importance that it seems desirable to give it farther consideration. Let us suppose that the total extension-in-phase for the kind of system considered to be divided into equal elements (i) F) which are very small but not infinitely small. Let us imagine an ensemble of systems distributed in this extension in a manner represented by the index of probability i;, which is an arbitrary function of the phase subject only to the re- striction expressed by equation (46) of Chapter I. We shall suppose the elements i> F to be so small that rf may in gen- eral be regarded as sensibly constant within any one of them at the initial moment. Let the path of a system be defined as the series of phases through which it passes. At the initial moment (^') a certain system is in an element of extension DV^. Subsequently, at the time ^", the same system is in the element D F". Other systems which were at first in i> F' will at the time t'' be in i> F", but not all, probably. The systems which were at first ia DV will at the time f' occupy an extension-in-phase exactly as large as at first. But it will probably be distributed among a very great number of the elements (-DF) into which we have divided the total extension-in-phase. If it is not so, we can generally take a later time at which it will be so. There wiU. be excep- tions to this for particular laws of motion, but we wiU con- fine ourselves to what may fairly be called the general case. Only a very small part of the systems initially in i> F' will be found in DF" at the time ^", and those which are found in DV" at that time were at the initial moment distributed among a very large number of elements D V. What is important for our purpose is the value of i;, the index of probability of phase in the element DV^^ at the time THROUGH LONG PERIODS OF TIME. 149 t^K In the part of D V" occupied by systems which at the time If were in DV* the value of i; will be the same as its value in i>F' at the time t\ which we shall call i;'. In the parts of D V^^ occupied by systems which at if were in ele- ments very near to i) F"' we may suppose the value of i; to vary little from 17'. We cannot assume this in regard to parts of DV" occupied by systems which at If were in elements remote from DV^. We want, therefore, some idea of the nature of the extension-in-phase occupied at f by the sys- tems which at V^ will occupy D V. Analytically, the prob- lem is identical with finding the extension occupied at t'^ by the systems which at If occupied DV^. Now the systems in D V which lie on the same path as the system first con- sidered, evidently arrived at DV at nearly the same time, and must have left D V at nearly the same time, and there- fore at if were in or near DV^. We may therefore take 7;' as the value for these systems. The same essentially is true of systems in BV which lie on paths very close to the path already considered. But with respect to paths passing through D V and D F"", but not so close to the first path, we cannot assume that the time required to pass from DV to DV'^ is nearly the same as for the first path. The difference of the times required may be small in comparison with <"-^', but as this interval can be as large as we choose, the difference of the times required in the different paths has no limit to its pos- sible value. Now if the case were one of statistical equilib- rium, the value of 17 would be constant in any path, and if aU the paths which pass through D V also pass through or near DV^y the value of rj throughout DV will vary little fi'om 7)'. But when the case is not one of statistical equilibrium, we cannot draw any such conclusion. The only conclusion which we can draw with respect to the phase at t' of the sys- tems which at t^' are in D V is that they are nearly on the same path. Now if we should make a new estimate of indices of prob- ability of phase at the time <", using for this purpose the elements DV] — that is, if we should divide the number of 150 MOTION OF SYSTEMS AND ENSEMBLES systems in D F", for example, by the total nimiber of systems, and also by the extension-in-phase of the element, and take the logarithm of the quotient, we would get a number which would be less than the average value of rf for the systems within DV" based on the distribution in phase at the time ^'.* Hence the average value of ij for the whole ensemble of systems based on the distribution at t^' wiU. be less than the average value based on the distribution at t^. We must not forget that there are exceptions to this gen- eral rule. These exceptions are in cases in which the laws of motion are such that systems having small differences of phase will continue always to have small differences of phase. It is to be observed that if the average index of probability in an ensemble may be said in some sense to have a less value at one time than at another, it is not necessarily priority in time which determines the greater average index. If a distribution, which is not one of statistical equilibrium, should be given for a time ^', and the distribution at an earlier time ^" should be defined as that given by the corresponding phases, if we increase the interval leaving t' fixed and taking t^' at an earlier and earlier date, the distribution at t'^ will in general approach a limiting distribution which is in statistical equilibrium. The determining difference in such cases is that between a definite distribution at a definite time and the limit of a varying dis- tribution when the moment considered is carried either forward or backward indefinitely, f But while the distinction of prior and subsequent events may be immaterial with respect to mathematical fictions, it is quite otherwise with respect to the events of the real world. It should not be forgotten, when our ensembles are chosen to illustrate the probabilities of events in the real world, that « See Chapter XI, Theorem IX. t One may compare the kinematical truism that when two points are moving with uniform velocities, (with the single exception of the case where the relative motion is zero,) their mutual distance at any definite time is less than for t=:oo,ort = — a. THROUGH LONG PERIODS OF TIME, 151 while the probabilities of subsequent events may often be determined from the probabilities of prior events, it is rarely the ease that probabilities of prior events can be determined from those of subsequent events, for we are rarely justified in excluding the consideration of the antecedent probability of the prior events. It is worthy of notice that to take a system at random from an ensemble at a date chosen at random from several given dates, t\ V\ etc., is practically the same thing as to take a sys- tem at random from the ensemble composed of all the systems of the given ensemble in their phases at the time t\ together with the same systems in their phases at the time t", etc. By Theorem VIII of Chapter XI this will give an ensemble in which the average index of probability will be less than in the given ensemble, except in the case when the distribution in the given ensemble is the same at the times t\ ^", etc. Consequently, any indefiniteness in the time in which we take a system at random from an ensemble has the practical effect of diminishing the average index of the ensemble from which the system may be supposed to be drawn, except when the given ensemble is in statistical equilibrium. CHAPTER Xm. EFFECT OP VARIOUS PROCESSES ON AN ENSEMBLE OP SYSTEMS. In the last chapter and m Chapter I we have considered the changes which take place in the course of time in an ensemble of isolated systems. Let us now proceed to consider the changes which will take place in an ensemble of systems under external influences. These external influences will be of two kinds, the variation of the coordinates which we have called external^ and the action of other ensembles of systems. The essential difference of the two kinds of influence consists in this, that the bodies to which the external coordinates relate are not distributed in phase, while in the case of interaction of the systems of two ensembles, we have to regard the feet that both are distributed in phase. To find the effect pro- duced on the ensemble with which we are principally con- cerned, we have therefore to consider single values of what we ha\e called external coordinates, but an infinity of values of the internal coordinates of any other ensemble with which there is interaction. Or, — to regard the subject from another point of view, — the action between an unspecified system of an ensemble and the bodies represented by the external coordinates, is the action between a system imperfectly determined with respect to phase and one which is perfectly determined; while the interaction between two unspecified systems belonging to different ensembles is the action between two systems both of which are imperfectly determined with respect to phase.* We shall suppose the ensembles which we consider to be distributed in phase in the manner described in Chapter I, and * In the development of the subject, we shaU find that this distinction corresponds to the distinction in thermodynamics between mechanical and thermal action. EFFECT OF VARIOUS PROCESSES. 158 represented by the notations of that chapter, especially by the index of probability of phase (t)^. There are therefore 2 n independent variations in the phases which constitute the ensembles considered. This excludes ensembles like the microcanonical, in which, as energy is constant, there are only 2 n — 1 independent variations of phase. This seems necessary for the purposes of a general discussion. For although we may imagine a microcanonical ensemble to have a permanent existence when isolated from external influences, the effect of such influences would generally be to destroy the imiformity of energy in the ensemble. Moreover, since the microcanonical ensemble may be regarded as a limiting case of such ensembles as are described in Chapter I, (and that in more than one way, as shown in Chapter X,) the exclusion is rather formal than real, since any properties which belong to the microcanonical ensemble could easily be derived from those of the ensembles of Chapter I, which in a certain sense may be regarded as representing the general case. Let us first consider the effect of variation of the external cobrdinates. We have already had occasion to regard these quantities as variable in the differentiation of certain equations relating to ensembles distributed according to certain laws called canonical or microcanonical. That variation of the external coordinates was, however, only carrying the atten- tion of the mind from an ensemble with certain values of the external coordinates, and distributed in phase according to some general law depending upon those values, to another ensemble with different values of the external coordinates, and with the distribution changed to conform to these new values. What we have now to consider is the effect which would actually result in the course of time in an ensemble of systems in which the external coordinates should be varied in any arbitrary manner. Let us suppose, in the first place, that these coordinates are varied abruptly at a given instant, being constant both before and after that instant. By the definition of the external coordinates it appears that this variation does not affect the phase of any system of the ensemble at the time 154 EFFECT OF VARIOUS PROCESSES when it takes place. Therefore it does not affect the index of probability of phase (rj') of any system, or the average value of the index (rj') at that time. And if these quantities are constant in time before the variation of the external cob'rdi- nates, and after that variation, their constancy in time is not interrupted by that variation. In fact, in the demonstration of the conservation of probability of phase in Chapter I, the variation of the external coordinates was not excluded. But a variation of the external cob'rdinates will in general disturb a previously existing state of statistical equilibrium. For, although it does not affect (at the first instant) the distribution-in-phase, it does affect the condition necessary for equilibrium. This condition, as we have seen in Chapter IV, is that the index of probability of phase shall be a function of phase which is constant in time for moving systems. Now a change in the external coordinates, by changing the forces which act on the systems, will change the nature of the functions of phase which are constant in time. Therefore, the distribution in phase which was one of statistical equi- librium for the old values of the external coordinates, will not be such for the new values. Now we have seen, in the last chapter, that when the dis- tribution-in-phase is not one of statistical equilibrium, an ensemble of systems may, and in general will, after a longer or shorter time, come to a state which may be regarded, if very small differences of phase are neglected, as one of statistical equilibrium, and in which consequently the average value of the index (^) is less than at first. It is evident, therefore, that a variation of the external coordinates, by disturbing a state of statistical equilibrium, may indirectly cause a diminu- tion, (in a certain sense at least,) of the value of rj. But if the change in the external coordinates is very small, the change in the distribution necessary for equilibrium will in general be correspondingly small. Hence, the original dis- tribution in phase, since it differs little from one which would be in statistical equilibrium with the new values of the ex- ternal coordinates, may be supposed to have a value of v ON AN ENSEMBLE OF SYSTEMS. 155 which differs by a small quantity of the second oider from the minimum value which characterizes the state of statistical equilibrium. And the diminution in the average index result- ing in the course of time from the very small change in the external coordinates, cannot exceed this small quantity of the second order. Hence also, if the change in the external coordinates of an ensemble initially in statistical equilibrium consists in suc- cessive very small changes separated by very long intervals of time in which the disturbance of statistical equilibrium be- comes sensibly effaced, the final diminution in the average index of probability will in general be negligible, although the total change in the external coordinates is large. The result will be the same if the change in the external coordinates takes place continuously but sufficiently slowly. Even in cases in which there is no tendency toward the restoration of statistical equilibrium in the lapse of time, a varia- tion of external coordinates which would cause, if it took place in a short time, a great disturbance of a previous state of equilibrium, may, if sufficiently distributed in time, produce no sensible disturbance of the statistical equilibrium. Thus, in the case of three degrees of freedom, let the systems be heavy points suspended by elastic massless cords, and let the ensemble be distributed in phase with a density proportioned to some function of the energy, and therefore in statistical equi- librium. For a change in the external coordinates, we may take a horizontal motion of the point of suspension. If this is moved a given distance, the resulting disturbance of the statistical equilibrium may evidently be diminished indefi- nitely by diminishing the velocity of the point of suspension. This will be true if the law of elasticity of the string is such that the period of vibration is independent of the energy, in which case there is no tendency in the course of time toward a state of statistical equilibrium, as well as in the more general case, in which there is a tendency toward statistical equilibrium. That something of this kind will be true in general, the following considerations wiU tend to show. 156 EFFECT OF VARIOUS PROCESSES We define a path as the series of phases through which a system passes in the course of time when the external co^ oidinates have fixed values. When the external coordinates are varied, paths are changed. The path of a phase is the path to which that phase belongs. With reference to any ensemble of systems we shall denote by T7\p the average value of the density-in-phase in a path. This implies that we have a measure for comparing different portions of the path. We shall suppose the time required to traverse any portion of a path to be its measure for the purpose of determining this average. With this understanding, let us suppose that a certain en- semble is in statistical equilibrium. In every element of extension-in-phase, therefore, the density-in-phase D is equal to its path-average lJ\p. Let a sudden small change be made in the external cob'rdinates. The statistical equilibrium will be disturbed and we shall no longer have D = 77]^ everywhere. This is not because D is changed, but because lJ\p is changed, the paths being changed. It is evident that if 2> > IJ]p in a part of a path, we shall have D < 77]^ in other parts of the same path. Now, if we should imagine a further change in the external coordinates of the same kind, we should expect it to produce an effect of the same kind. But the manner in which the second effect will be superposed on the first will be different, according as it occurs immediately after the first change or after an interval of time. If it occurs immediately after the first change, then in any element of phase in which the first change produced a positive value of D - TJ\p the second change will add a positive value to the first positive value, and where D - 2?lp was negative, the second change will add a negative value to the first negative value. But if we wait a sufficient time before making the second change in the external coordinates, so that systems have passed from elements of phase in which D - 2?]^ was origi- nally positive to elements in which it was originally negative, and vice versa, (the systems carrying with them the values ON AN ENSEMBLE OF SYSTEMS. 157 of D - U\p ,) the positive values of D - U],, caused by the second change will be in part superposed on negative values due to the first change, and vice versa. The disturbance of statistical equilibrium, tiieiefore, pro- duced by a given change in the values of the external co- ordinates may be very much diminished by dividing the change into two parts separated by a sufficient interval of time, and a sufficient interval of time for this purpose is one in which the phases of the individual systems are entirely unlike the first, so that any individual system is differently affected by the change, although the whole ensemble is af- fected in nearly the same way. Since there is no limit to the diminution of the disturbance of equilibrium by division of the change in the external cob'rdinates, we may suppose as a general rule that by diminishing the velocity of the changes in the external coordinates, a given change may be made to produce a very small disturbance of statistical equilibrium. If we write ^' for the value of the average index of probability before the variation of the external coordinates, and ^" for the value after this variation, we shall have in any case as the simple result of the variation of the external codrdi- nates. This may be compared with the thermodynamic the- orem that the entropy of a body cannot be diminished by mechanical (as distinguished from thermal) action.* If we have (approximate) statistical equilibrium between the times if and t" (corresponding to rj' and ^"), we shall have approximately ? = V? which may be compared with the thermodynamic theorem that the entropy of a body is not (sensibly) affected by mechanical action, during which the body is at each instant (sensibly) in a state of thermodynamic equilibrium. Approximate statistical equilibrium may usually be attained * The correspondences to which the reader's attention is called are between — 1} and entropy, and between 6 and temperature. 158 EFFECT OF VARIOUS PROCESSES by a sufiBciently slow variation of the external cob'rdinates, just as approximate thermodynamic equilibrium may usually be attained by sufficient slowness in the mechanical operations to which the body is subject. We now pass to the consideration of the effect on an en- semble of systems which is produced by the action of other ensembles with which it is brought into djmamical connec- tion. In a previous chapter * we have imagined a dynamical connection arbitrarily created between the systems of two ensembles. We shall now regard the action between the systems of the two ensembles as a result of the variation of the external cobrdinates, which causes such variations of the internal coordinates as to bring the systems of the two ensembles within the range of each other's action. Initially, we suppose that we have two separate ensembles of systems, U^ and U^* T^® numbers of degrees of freedom of the systems in the two ensembles wiU be denoted by n^ and Wg respectively, and the probability-coefficients by e"^ and e"^. Now we may regard any system of the first ensemble com- bined with any system of the second as forming a single system of n^ + n^ degrees of freedom. Let us consider the ensemble (^12) obtained by thus combining each system of the first ensemble with each of the second. At the initial moment, which may be specified by a single accent, the probability-coefficient of any phase of the combined systems is evidentiy the product of the probability-coefficients of the phases of which it is made up. This may be expressed by the equation, gW = e"^' e"^', (455) or ,yi/ = ,;/ + ,;/, (456) which gives ^2' = vi + V^* (^"0 The forces tending to vary the internal coordinates of the combined systems, together with those exerted by either system upon the bodies represented by the coordinates called * See Chapter IV, page 87. ON AN ENSEMBLE OF SYSTEMS. 159 external, may be derived from a single force-function, which, taken negatively, we shall call the potential energy of the combined systems and denote by e^g. But we suppose that initially none of the systems of the two ensembles Hi and E^ come within range of each other's action, so that the potential energy of the combined system falls into two parts relating separately to the systems which are combined. The same is obviously true of the kinetic energy of the combined compound system, and therefore of its total energy. This may be expressed by the equation €i,' = €/ + €/, (458) which gives €12' = «/ + €2'- (459) Let us now suppose that in the course of time, owing to the motion of the bodies represented by the codrdinates called external, the forces acting on the systems and consequently their positions are so altered, that the systems of the ensembles E^ and E^ are brought within range of each other's action, and after such mutual influence has lasted for a time, by a further change in the external cob'rdinates, perhaps a return to their original values, the systems of the two original en- sembles are brought again out of range of each other's action. Finally, then, at a time specified by double accents, we shall have as at first ii2" = ii" + i2". (460) But for the indices of probability we must write * 171 T^ 172 ^ 1712 • (461) The considerations adduced in the last chapter show that it is safe to write W' < W. (462) We have therefore ^1" + ^" < vi! + ^Jy (463) which may be compared with the thermodynamic theorem that * See Chapter XI, Theorem VIL 160 EFFECT OF VARIOUS PROCESSES the thermal contact of two bodies may increase but cannot diminish the sum of their entropies. Let us especially consider the case in which the two original ensembles were both canonically distributed in phase with the respective moduli ©^ and ©j. We have then, by Theorem III of Chapter XI, ^x' + '^<^"+g (464) V.' + |;'^^" + ^ (466) Whence with (468) we have S^ + ^^>0. (467) If we write W for the average work done by the combined systems on the external bodies, we have by the principle of the conservation of energy W = €t^' - €i/' = €/ - ei" + €,' - ^". (468) Now if Tfis negligible, we have i/' - ii' = - (€7' - ^0 (469) and (467) shows that the ensemble which has the greater modulus must lose energy. This result may be compared to the thermodynamic principle, that when two bodies of differ- ent temperatures are brought together, that which has the higher temperature wiU lose eneigy. Let us next suppose that the ensemble U^ is originally canonically distributed with the modulus ©g , but leave the distribution of the other arbitrary. We have, to determine the result of a similar process, Vi" + V," ^Vi' + vJ ON AN ENSEMBLE OF SYSTEMS. 161 Hence vi" + |' ^ V + ^ (470) which may be written 0. ^' - ^'' > '-i—^ (471) This may be compared with the thermodynamic principle that when a body (which need not be in thermal equilibrium) is brought into thermal contact with another of a given tempera- ture, the increase of entropy of the first cannot be less (alge- braically) than the loss_of heat by the second divided by its temperature. Where W is negligible, we may write V' + ^^V + I (472) Now, by Theorem III of Chapter XI, the quantity ^ + 1 (473) has a minimum value when the ensemble to which Vi and e^ relate is distributed canonically with the modulus ©g* ^^ ^® ensemble had originally this distribution, the sign < in (472) would be impossible. In fact, in this case, it would be easy to show that the preceding formulae on which (472) is founded would all have the sign = . But when the two ensembles are not both originally distributed canonically with the same modulus, the formulae indicate that the quantity (473) may be diminished by bringing the ensemble to which €i and ly^ relate into connection with another which is canonically dis- tributed with modulus ©g, and therefore, that by repeated operations of this kind the ensemble of which the original dis^ tribution was entirely arbitrary might be brought approxi- mately into a state of canonical distribution with the modulus 02- We may compare this with the thermodynamic principle^ that a body of which the original thermal state may be entirely arbitrary, may be brought approximately into a state of ther- mal equilibrium with any given temperature by repeated coiir nections with other bodies of that temperature. 11 162 EFFECT OF VARIOUS PROCESSES Let us now suppose that we have a certain number of ensembles, HqjU^^ U^^ etc., distributed canonically with the respective moduli ©q , ©j , ©2 » ^^* ^7 variation of the exter- nal coordinates of the ensemble Uq , let it be brought into connection with JEr^ , and then let the connection be broken. Let it then be brought into connection with H^ , and then let that connection be broken. Let this process be continued with respect to the remaining ensembles. We do not make the assumption, as in some cases before, that the work connected with the variation of the external co(5rdinates is a negligible quantity. On the contrary, we wish especially to consider the case in which it is large. In the final state of the ensem- ble Uq , let us suppose that the external coordinates have been brought back to their original values, and that the average energy (Jq) is the same as at first. In our usual notations, using one and two accents to dis- tinguish original and final values, we get by repeated applica- tions of the principle expressed in (463) Vo' + vi'+ V + etc. > ^0" + ?i" + W' + etc. (474) But by Theorem III of Chapter XI, W' + IJ^V + I'. (475) ^" + 1^ > ^' + |'» (476) ^" + |;>^.' + |. (477) etc. Hence ^ + !l!: + "!?!! + etc. > |^ + ^ + ^ + etc. (478) or, since V = €©", >^i-7r-^ + ^^-77-^ + etc. (479) If we write Tffor the average work done on the bodies repre- sented by the external coordinates, we have ON AN ENSEMBLE OF SYSTEMS. 163 €i' - Ci" + €a' - €2'' + etc. = W. (480) If Hq, jE^^, and ^^ ^^® ^^® ^^^Y ensembles, we have (BL — ®o - ^ ^ ^ ^\^ («.' - ci"). (481) It wiU be observed that ^e relations expressed m the last three formulae between Wj e^ — Cj", €3' — Cg", etc., and ©j, 02» etc. are precisely those which hold in a Camot's cycle for the work obtained, the energy lost by the several bodies which serve as heaters or coolers, and their initial temperatures. It will not escape the reader's notice, that while from one point of view the operations which are here described are quite beyond our powers of actual performance, on account of the impossibility of handling the immense number of systems which are involved, yet from another point of view the opera- tions described are the most simple and accurate means of representing what actually takes place in our simplest experi- ments in thermodynamics. The states of the bodies which we handle are certainly not known to us exactly. What we know about a body can generally be described most accurately and most simply by saying that it is one taken at random from a great number (ensemble) of bodies which are com- pletely described. If we bring it into connection with another body concerning which we have a similar limited knowledge, the state of the two bodies is properly described as that of a pair of bodies taken from a great number (ensemble) of pairs which are formed by combining each body of the first en- semble with each of the second. Again, when we bring one body into thermal contact with another, for example, in a Camot's cycle, when we bring a mass of fluid into thermal contact with some other body from which we wish it to receive heat, we may do it by moving the vessel containing the fluid. This motion is mathematically expressed by the variation of the coordinates which determine the position of the vessel. We allow ourselves for the pur- poses of a theoretical discussion to suppose that the walls of this vessel are incapable of absorbing heat from the fluid. 164 EFFECT OF VARIOUS PROCESSES. Yet while we exclude the kind of action which we call ther- mal between the fluid and the containing vessel, we allow the kind which we call work in the narrower sense, which takes place when the volume of the fluid is changed by the motion of a piston. This agrees with what we have supposed in regard to the external coordinates, which we may vary in any arbitrary manner, and are in this entirely unlike the co- ordinates of the second ensemble with which we bring the first into connection. When heat passes in any thermodynamic experiment between the fluid principally considered and some other body, it is actually absorbed and given out by the walls of the vessel, which will retain a varying quantity. This is, however, a disturbing circumstance, which we suppose in some way made negligible, and actually neglect in a theoretical discussion. In our case, we suppose the walls incapable of absorbing en- ergy, except through the motion of the external coordinates, but that they allow the systems which they contain to act directly on one another. Properties of this kind are mathe- matically expressed by supposing that in the vicinity of a certain surface, the position of which is determined by certain (external) coCrtiinates, particles belonging to the system in question experience a repulsion from the surface increasing so »pidly wiaTneames, Ja.e ,uAc that .0 ir^U, expeLi- ture of energy would be required to carry them through it. It is evident that two systems might be separated by a surface or surfaces exerting the proper forces, and yet approach each other closely enough to exert mechanical action on each other. CHAPTER XIV. DISCUSSION OF THERMODYNAMIC ANALOGIES. If we wish to find in rational mechanics an a priori founda- tion for the principles of thermodynamics, we must seek mechanical definitions of temperature and entropy. The quantities thus defined must satisfy (under conditions and with limitations which again must be specified in the language of mechanics) the differential equation da = Tdrf — Ai doi — A^ da^ — etc., {^^) where €, T^ and 17 denote the energy, temperature, and entropy of the system considered, and A^da^^ etc., the mechanical work (in the narrower sense in which the term is used in thermo- dynamics, t. ^., with exclusion of thennal action) done upon external bodies. This implies that we are able to distinguish in mechanical terms the thermal action of one system on another from that which we call mechanical in the narrower sense, if not indeed in every case in which the two may be combined, at least so as to specify cases of thermal action and cases of mechanical action. Such a differential equation moreover implies a finite equa- tion between €, 17, and a^, a^, etc., which may be regarded as fundamental in regard to those properties of the system which we call thermodynamic, or which may be called so from analogy. This fundamental thermodynamic equation is de- termined by the fundamental mechanical equation which expresses the energy of the system as function of its mo- menta and coordinates with those external coordinates (a^, Oj, etc.) which appear in the differential expression of the work done on external bodies. We have to show the mathematical operations by which the fundamental thermodynamic equation, 166 THERMODYNAMIC ANALOGIES. which in general is an equation of few variables, is derived from the fundamental mechanical equation, which in the case of the bodies of nature is one of an enormous number of variables. We have also to enunciate in mechanical terms, and to prove, what we call the tendency of heat to pass from a sys- tem of higher temperature to one of lower, and to show that this tendency vanishes with respect to systems of the same temperature. At least, we have to show by a priori reasoning that for such systems as the material bodies which nature presents to us, these relations hold with such approximation that they are sensibly true for human faculties of observation. This indeed is all that is really necessary to establish the science of thermodynamics on an a priori basis. Yet we will naturally desire to find the exact expression of those principles of which the laws of thermodynamics are the approximate expression. A very little study of the statistical properties of conservative systems of a finite number of degrees of freedom is sufficient to make it appear, more or less distinctly, that the general laws of thermodynamics are the limit toward which the exact laws of such systems approximate, when their number of degrees of freedom is indefinitely increased. And the problem of finding the exact relations, as distinguished from the ap- proximate, for systems of a great number of degrees of free- dom, is practically the same as that of finding the relations which hold for any number of degrees of freedom, as distin- guished from those which have been established on an em- pirical basis for systems of a great number of degrees of freedom. The enunciation and proof of these exact laws, for systems of any finite number of degrees of freedom, has been a princi- pal object of the preceding discussion. But it should be dis- tinctly stated that, if the results obtained when the numbers of degrees of freedom are enormous coincide sensibly with the general laws of thermodynamics, however interesting and significant this coincidence may be, we are still far from THERMODYNAMIC ANALOGIES. 167 having explained the phenomena of nature with respect to these laws. For, as compared with the case of nature, the systems which we have considered are of an ideal simplicity. Although our only assumption is that we are considering conservative systems of a finite number of degrees of freedom, it would seem that this is assuming far too much, so far as the bodies of nature are concerned. The phenomena of radiant heat, which certainly should not be neglected in any complete system of thermodynamics, and the electrical phenomena associated with the combination of atoms, seem to show that the hypothesis of systems of a finite number of degrees of freedom is inadequate for the explanation of the properties of bodies. Nor do the results of such assumptions in every detail appear to agree with experience. We should expect, for example, that a diatomic gas, so far as it could be treated independently of the phenomena of radiation, or of any sort of electrical manifestations, would have six degrees of freedom for each molecule. But the behavior of such a gas seems to indicate not more than five. But although these difficulties, long recognized by physi- cists,* seem to prevent, in the present state of science, any satisfactory explanation of the phenomena of thermodynamics as presented to us in nature, the ideal case of systems of a finite number of degrees of freedom remains as a subject which is certainly not devoid of a theoretical interest, and which may serve to point the way to the solution of the far more difficult problems presented to us by nature. And if the study of the statistical properties of such systems gives us an exact expression of laws which in the limiting case take the form of the received laws of thermodynamics, its interest is so much the greater. Now we have defined what we have called the modulus (0) of an ensemble of systems canonically distributed in phase, and what we have called the index of probability (i;) of any phase in such an ensemble. It has been shown that between * See Boltzmann, Sitzb. der Wiener Akad., Bd. LXIH., S. 418, (1871). 168 THERMODYNAMIC ANALOGIES. the modulus (6), the external coordinates (a^ , etc.), and the average values in the ensemble of the energy (e), the index of probability (17), and the external forces (J.^, etc.) exerted by the systems, the following differential equation will hold : cilj = — © c?^ — JTi doi — J", cfoj — etc. (483) This equation, if we neglect the sign of averages, is identical in form with the thermodynamic equation (482), the modulus (6) corresponding to temperature, and the index of probabil- ity of phase with its sign reversed corresponding to entropy.* We have also shown that the average square of the anoma- lies of €, that is, of the deviations of the individual values from the average, is in general of the same order of magnitude as the reciprocal of the niunber of degrees of freedom, and there- fore to human observation the individual values are indistin- guishable from the average values when the number of degrees of freedom is very great-f In this case also the anomalies of 1; are practically insensible. The same is true of the anomalies of the external forces {A^ , etc.), so far as these are the result of the anomalies of energy, so that when these forces are sensibly determined by the energy and the external coordinates, and the niunber of degrees of freedom is very great, the anomalies of these forces are insensible. The mathematical operations by which the finite equation between €, 17, and a^ , etc., is deduced from that which gives the energy (c) of a system in terms of the momenta (p^ . . . .p^ and coordinates both internal (g'l • • • ?») and external (a^ , etc.), are indicated by the equation e ^ =z j , , . j e ^dqi . . . dq^tdpi . . . dp^f (484) phases where ^ = ©^ + i. We have also shown that when systems of different ensem- bles are brought into conditions analogous to thermal contact, the average result is a passage of energy from the ensemble » See Chapter IV, pages 44, 46. t See Chapter VII, pages 7^-75. THERMODYNAMIC ANALOGIES. 169 of the greater modulus to that of the less, * or in case of equal moduli, that we have a condition of statistical equilibrium in regard to the distribution of energy.f Propositions have also been demonstrated analogous to those in thermodynamics relating to a Camot's cycle,J or to the tendency of entropy to increase, § especially when bodies of different temperature are brought into contact. || We have thus precisely defined quantities, and rigorously demonstrated propositions, which hold for any number of degrees of freedom, and which, when the number of degrees of freedom (n) is enormously great, would appear to human faculties as the quantities and propositions of empirical ther- modynamics. It is evident, however, that there may be more than one quantity defined for finite values of w, which approach the same limit, when n is increased indefinitely, and more than one proposition relating to finite values of n, which approach the same limiting form for w = oo. There may be therefore, and there are, other quantities which may be thought to have some claim to be regarded as temperature and entropy with respect to systems of a finite number of degrees of freedom. The definitions and propositions which we have been con- sidering relate essentially to what we have called a canonical ensemble of systems. This may appear a less natural and simple conception than what we have called a microcanonical ensemble of systems, in which all have the same energy, and which in many cases represents simply the time-ensembhy or en^eMMe of pLas toV which /.igk sytem pa»e. in the course of time. It may therefore seem desimble to find definitions and propositions relating to these microcanonical ensembles, which shall correspond to what in thermodynamics are based on experience. Now the differential equation d€ = 6""^ Vd log r- 3rL doi — 3^1, rfoj — etc., (486) * See Chapter XIII, page 160. t See Chapter IV, pages 86-37. } See Chapter XIII, pages 162, 163. § See Chapter XII, pages 148-151. 11 See Chapter XIII, page 159. 170 THERMODYNAMIC ANALOGIES, which has been demonstrated in Chapter X, and which relates to a microcanonical ensemble,^ denotmg the average value of -4i in such an ensemble, corresponds precisely to the thermody- namic equation, except for the sign of average applied to the external forces. But as these forces are not entirely deter- mined by the energy with the external coordinates, the use of average values is entirely germane to the subject, and affords the readiest means of getting perf ectiy determmed quantities. These averages, which are taken for a microcanonical ensemble, may seem from some points of view a more simple and natural conception than those which relate to a canonical ensemble. Moreover, the energy, and the quantity corresponding to en- tropy, are free from the sign of average in this equation. The quantity in the equation which corresponds to entropy is log FJ the quantity V being defined as the extension-in- phase within which the energy is less than a certain limiting value (e). This is certainly a more simple conception than the average value in a canonical ensemble of the index of probabil- ity of phase. Log V has the property that when it is constant die = — 371f da-x — !Z7|f da^ + etc., (486) which closely corresponds to the thermodynamic property of entropy, that when it is constant cfc = — -4i da^ — A^ da^ + etc. (^7) The quantity in the equation which corresponds to tem- perature is €~* F", or defd log V. In a canonical ensemble, the average value of this quantity is equal to the modulus, as has been shown by different methods in Chapters IX and X. In Chapter X it has also been shown that if the systems of a microcanonical ensemble consist of parts with separate energies, the average value of e"^ Ff or any part is equal to its average value for any other part, and to the uniform value of the same expression for the whole ensemble. This corre- sponds to the theorem in the theory of heat that in case of thermal equilibrium the temperatures of the parts of a body are equal to one another and to that of the whole body. THERMODYNAMIC ANALOGIES. 171 Since the energies of the parts of a body cannot be supposed to remain absolutely constant, even where this is the case with respect to the whole body, it is evident that if we regard the temperature as a function of the energy, the taking of average or of probable values, or some other statistical process, must be used with reference to the parts, in order to get a perfectly definite value corresponding to the notion of tem- perature. It is worthy of notice in this connection that the average value of the kinetic energy, either in a microcanonical en- semble, or in a canonical, divided by one half the number of degrees of freedom, is equal to e""* FJ or to its average value, and that this is true not only of the whole system which is distributed either microcanonically or canonically, but also of any part, although the corresponding theorem relating to temperature hardly belongs to empirical thermodjmamics, since neither the (inner) kinetic energy of a body, nor its number of degrees of freedom is immediately cognizable to our facul- ties, and we meet the gravest difficulties when we endeavor to apply the theorem to the theory of gases, except in the simplest case, that of the gases known as monatomic. But the correspondence between e-^ V or de\d log V and temperature is imperfect. If two isolated systems have such energies that dei de^ dflog Vx^ dlog Fa' and the two systems are regarded as combined to form a third system with energy Cll = €l + €l» we shall not have in general d€i2 d€i d€2 rflog Fia "" dlog Vi " dlog V^ as analogy with temperature would require. In fact, we have seen that d log r, ia "" d log Filfia "" d\o%v\^^ 172 THERMODYNAMIC ANALOGIES. where the second and third members of the equation denote average values in an ensemble in which the compound system is microcanonically distributed in phase. Let us suppose the two original systems to be identical in nataie. Then The equation in question would require that dei aei dlog Fi""rflog ViV^' « t. e.9 that we get the same result, whether we take the value of dejdlog V^ determined for the average value of c^ in the ensemble, or take the average value of de^/dlog Vy This will be the case where de^/dlog F^ is a linear function of €y Evidently this does not constitute the most general case. Therefore the equation in question cannot be true in general. It is true, however, in some very important particular cases, as when the energy is a quadratic function of the ^s and q% or of the p's alone.* When the equation holds, the case is anal- ogous to that of bodies in thermodynamics for which the specific heat for constant volume is constant. Another quantity which is closely related to temperature is d 2, the average value of dt^jde is 1/6, and that the most common value of the energy in the ensemble is that for which d^jde = 1/0. The first of these properties may be compared with that of de/dlog V, which has been seen to have the average value in a canonical ensemble, without restriction in regard to the number of degrees of freedom. With respect to microcanonical ensembles also, d(f)/d€ has a property similar to what has been mentioned with respect to de/d log V. That is, if a system microcanonically distributed in phase consists of two parts with separate energies, and each * This last case is important on account of its relation to the theory of gases, although it must in strictness be regarded as a limit of possible cases, rather than as a case which is itself possible. THERMODYNAMIC ANALOGIES, 173 with more than two degrees of freedom, the average values in the ensemble of d/de for the two parts are equal to one another and to the value of same expression for the whole. In our usual notations o4\ d€i 3^ Km ^^ if »i > 2, and tIj > 2. This analogy with temperature has the same incompleteness which was noticed with respect to de/dlog Vy viz., if two sys- tems have such energies (e^ and e,) that and they are combined to form a third system with energy we shall not have in general i (J^j d€i2 dei d€2 ' Thus, if the energy is a quadratic function of the jp's and q% we have * di fii — 1 d2 Wj — 1 dtfiii _ ^18 — 1 9ii + n^ — 1 *12 ~ €i2 €i + €a where t^ 9 ^29 ^129 are the numbers of degrees of freedom of the separate and combined systems. But di d2 ^ + ^2 — 2 cfei dcf €1 + C2 If the energy is a quadratic function of the jt?'s alone, the case would be the same except that we should have J w^ , i nj , J Wjj , instead of n^ , 7^, n^s* In these particular cases, the analogy * See foot-note on page 93. We have here made the least value of the energy consistent with the values of the external coordinates zero instead of fa, as is evidently allowable when the external coordinates are supposed invariable. 174 THERMODYNAMIC ANALOGIES. between de/dlog V and temperature would be complete, as has akeady been remarked. We should have d€i _€i^ Cfe, 6, dlogVi fii dlogVf «,' Cfelf _ ^12 _ dci __ lde are its properties relating to most probable values of energy. If a system having two parts with separate energies and each with more than two degrees of freedom is microcanonically distributed in phase, the most probable division of energy between the parts, in a system taken at random from the ensemble, satisfies the equation which corresponds to the thermodynamic theorem that the distribution of energy between the parts of a system, in case of thermal equilibrium, is such that the temperatures of the parts are equal. To prove the theorem, we observe that the fractional part of the whole number of systems which have the energy of one part (e^) between the limits e^ and e^" is expressed by ■/' —011 1 0i+0a , e I e a€i, where the variables are connected by the equation Cj + €2 = constant = €12. The greatest value of this expression, for a constant infinitesi- mal value of the difference e^" — ej', determines a value of e^ , which we may call its most probable value. This depends on the greatest possible value of i + ^2- Now if n^ > 2, and nj > 2, we shall have <^j = — oo for the least possible value of THERMODYNAMIC ANALOGIES. 175 €i, and <^2 = — 00 for the least possible value of €3. Between these limits ^^ and ^ will be finite and continuous. Hence ^ 4- ^2 ^'^ have a maximum satisfying the equation (488). But if Tij ^ 2, or Wj ^ 2, d(f)Jd€i or rf<^2/^^2 ^^7 '^ nega- tive, or zero, for all values of e^ or €2, and can hardly be regarded as having properties analogous to temperature. It is also worthy of notice that if a system which is micro- canonically distributed in phase has three parts with separate energies, and each with more than two degrees of freedom, the most probable division of energy between these parts satisfies the equation nlde as corresponding to the reciprocal of temperature, or, in other words, d€/d(f> as corresponding to temperature, (f) will correspond to entropy. It has been defined as log {d Vide). In the considerations on which its definition is founded, it is therefore very similar to log V. We have seen that dtp/d log V approaches the value unity when n is very great.* To form a differential equation on the model of the thermo- dynamic equation (482), in which dejdif) shall take the place of temperature, and ^ of entropy, we may write d;^= fdc + gda, + gda, + etc (490) With respect to the differential coeflScients in the last equa- tion, which corresponds exactly to (482) solved with respect to rfiy, we have seen that their average values in a canonical ensemble are equal to 1/0, and the averages of -^j/Q, A^/S^ etc.f We have also seen that deldcf) (or d^jde) has relations to the most probable values of energy in parts of a microca- nonical ensemble. That (del da^^p^a^ etc., have properties somewhat analogous, may be shown as follows. In a physical experiment, we measure a force by balancing it against another. If we should ask what force applied to in- crease or diminish a^ would balance the action of the systems, it would be one which varies with the different systems. But we may ask what single force will make a given value of a^ the most probable, and we shall find that under certain condi- tions {de/da^(p, a represents that force. * See Chapter X, pages 120, 121. t See Chapter IX, equations (321), (327). THERMODYNAMIC ANALOGIES. 177 To make the problem definite, let us consider a system con- sisting of the original system together with another having the coordinates a^ , aj , etc., and forces -4^', A,^^ etc., tending to increase those cotjrdinates. These are in addition to the forces -^1, -^2, etc., exerted by the original system, and are de- rived from a force-function (— e^') by the equations M-i — — 3 — , xx% — — — f etc. ddi dUi For the energy of the whole system we may write E = € + €,' + ^mj ai^ + ^m, aj* + etc., and for the extension-in-phase of the whole system within any limits J ... I dpi • . . d^f^ doi rriri da,i da^ m^ da2 • • • or j . • > j e^de dai mi doi da^ m^ da^ • • • , or again j , . . j e^dE doi mi doi da^ m^ da2 . • . , since de = (2E, when a^, ai, a2, Os, etc., are constant. If the limits are expressed by £ and E + dEy a^ and a^ + da^ , a^ and a^ + (2a^ , etc., the integral reduces to 6^ dE, daimi da^ da^m^ da^ • • • The values of a^ , a^ , a^ , a2 , etc., which make this expression a maximum for constant values of the energy of the whole system and of the differentials dEy da^ , da^j etc., are what may be called the most probable values of a^ , a^ , etc., in an ensem- ble in which the whole system is distributed microcanonically. To determine these values we have de* = 0, when d(€ + c,' + i m fli* 4- ^ mj a," + etc.) = 0. That is, d = 0, 12 178 THERMODYNAMIC ANALOGIES. when (-77 ) d^<^ + ( -— J (ia, — -4/ rfoi + etc + wij d^ dii + etc. = 0. This requires ai = 0, a, = 0, etc., This shows that for any given values of E, a^, o^, etc. f ^ j , f -j^ j , etc., represent the forces (in the gen- eralized sense) which the external bodies would have to exert to make these values of a^, aj, etc., the most probable under the conditions specified. When the differences of the external forces which are exerted by the different systems are negli- gible, — (de/da^^^a^ etc., represent these forces. It is certainly in the quantities relating to a canonical ensemble, i, 0, ^, A^y etc., a^, etc., that we find the most complete correspondence with the quantities of the thermody- namic equation (482). Yet the conception itself of the canon- ical ensemble may seem to some artificial, and hardly germane to a natural exposition of the subject; and the quantities '' d^' i«g ^' ^^' «**'•' «!' ^^- °^^' ^' ^' (^),.„' etc., a^, etc., which are closely related to ensembles of constant energy, and to average and most probable values in such ensembles, and most of which are defined without reference to any ensemble, may appear the most natural analogues of the thermodynamic quantities. In regard to the naturalness of seeking analogies with the thermodynamic behavior of bodies in canonical or microca- nonical ensembles of systems, much will depend upon how we approach the subject, especially upon the question whether we regard energy or temperature as an independent variable. It is very natural to take energy for an independent variable rather than temperature, because ordinary mechanics furnishes us with a perfectly defined conception of energy, whereas the idea of something relating to a mechanical system and corre- THERMODYNAMIC ANALOGIES, 179 spending to temperature is a notion but vaguely defined. Now if the state of a system is given by its energy and the external cob'rdinates, it is incompletely defined, although its partial defi- nition is perfectly clear as far as it goes. The ensemble of phases microcanonicaUy distributed, with the given values of the energy and the external coordinates, will represent the im- perfectly defined system better than any other ensemble or single phase. When we approach the subject from this side, our theorems wiU naturally relate to average values, or most probable values, in such ensembles. In this case, the choice between the variables of (485) or of (489) will be determined partly by the relative importance which is attached to average and probable values. It would seem that in general average values are the most important, and that they lend themselves better to analytical transformations. This consideration would give the preference to the system of variables in which log V is the analogue of entropy. Moreover, if we make ^ the analogue of entropy, we are embarrassed by the necessity of making numerous exceptions for systems of one or two degrees of freedom. On the other hand, the definition of ^ may be regarded as a little more simple than that of log Vy and if our choice is deter- mined by the simphcity of the definitions of the analogues of entropy and temperature, it would seem that the <^ system should have the preference. In our definition of these quanti- ties, Fwas defined first, and e* derived from Fby differen- tiation. This gives the relation of the quantities in the most simple analytical form. Yet so far as the notions are con- cerned, it is perhaps more natural to regard Fas derived from e* by integration. At all events, e* may be defined inde- pendently of F, and its definition may be regarded as more simple as not requiring the determination of the zero from which F is measured, which sometimes involves questions of a delicate nature. In fact, the quantity «* may exist, when the definition of F becomes illusory for practical pur- poses, as the integral by which it is determined becomes infinite. The case is entirely different, when we regard the tempera- 180 THERMODYNAMIC ANALOGIES. tuie as an independent variable, and we have to consider a system which b described as having a certain temperature and certain values for the external coordinates. Here also the state of the system is not completely defined, and will be better represented by an ensemble of phases than by any single phase. What is the nature of such an ensemble as wiSl best represent the imperfectly defined state ? When we wish to give a bodya certain temperature, we place it in a bath of the proper temperature, and when we regard what we caU thermal equilibrium a^ established, we say that the body has the same temperature as the bath. Per- haps we place a second body of standard character, which we call a thermometer, in the bath, and say that the first body, the bath, and the thermometer, have all the same temperature. But the body under such circumstances, as well as the bath, and the thermometer, even if they were entirely isolated from external influences (which it is convenient to suppose in a theoretical discussion), would be continually changing in phase, and in energy as well as in other respects, although our means of observation are not fine enough to perceive these variations. The series of phases through which the whole system runs in the course of time may not be entirely determined by the energy, but may depend on the initial phase in other respects. In such cases the ensemble obtained by the microcanonical distribution of the whole system, which includes aU possible time-ensembles combined in the proportion which seems least arbitrary, wiU represent better than any one time-ensemWe the effect of the bath. Indeed a single time-ensemble, when it is not also a microcanonical ensemble, is too ill-defined a notion to serve the purposes of a general discussion. We will therefore direct our attention, when we suppose the body placed in a bath, to the microcanonical ensemble of phases thus obtained. If we now suppose the quantity of the substance forming the bath to be increased, the anomalies of the separate ener- gies of the body and of the thermometer in the microcanonical THERMODYNAMIC ANALOGIES. 181 ensemble will be increased, but not without limit. The anom- alies of the energy of the bath, considered in comparison with its whole energy, diminish indefinitely as the quantity of the bath is increased, and become in a sense negligible, when the quantity of the bath is sufficiently increased* The ensemble of phases of the body, and of the thermometer, approach a standard form as the quantity of the bath is in- definitely increased. This limiting form is easily shown to be what we have described as the canonical distribution. Let us write € for the energy of the whole system consisting of the body first mentioned, the bath, and the thermometer (if any), and let us first suppose this system to be distributed canonically with the modulus 0. We have by (205) d& and since ^p = o d® 2 de^ If we write Ae for the anomaly of mean square, we have (AC)« =(€-€)«. d% If we set A© = "^Ac, de A 6 wiU represent approximately the increase of which would produce an increase in the average value of the energy equal to its anomaly of mean square. Now these equations give 20»d!L which shows that we may diminish Ae indefinitely by increas- ing the quantity of the bath. Now our canonical ensemble consists of an infinity of micro- canonical ensembles, which diiffer only in consequence of the different values of the energy which is constant in each. If we consider separately the phases of the first body which 182 THERMODYNAMIC ANALOGIES. occur in the canonical ensemble of the whole system, these phases will form a canonical ensemble of the same modulus. This canonical ensemble of phases of the first body will con- sist of parts which belong to the different microcanonical ensembles into which the canonical ensemble of the whole Bystem is (iivided. Let us now imagine that the modulus of the principal ca- nonical ensemble is increased by 2 A0, and its average energy by 2 A 6. The modulus of the canonical ensemble of the phases of the first body considered separately will be increased by 2 A0. We may regard the infinity of microcanonical en- sembles into which we have divided the principal canonical ensemble as each having its energy increased by 2A6. Let us see how the ensembles of phases of the first body con- tained in these microcanonical ensembles are affected. We may assmne that they will all be affected in about the same way, as all the differences which come into account may be treated as small. Therefore, the canonical ensemble formed by taking them together wiU also be affected in the same way. But we know how this is affected. It is by the increase of its modulus by 2A@, a quantity which vanishes when the quantity of the bath is indefinitely increased. Li the case of an infinite bath, therefore, the increase of the energy of one of the microcanonical ensembles by 2A6, pro- duces a vanishing effect on the distribution in energy of the phases of the first body which it contains. But 2A€ is more than the average difference of energy between the micro- canonical ensembles. The distribution in energy of these phases is therefore the same in the different microcanonical ensembles, and must therefore be canonical, like that of the ensemble which they form when taken together.* * In order to appreciate the above reasoning, it should be understood that the differences of energy which occur in the canonical ensemble of phases of the first body are not here regarded as vanishing quantities. To fix one's ideas, one may imagine that he has the fineness of perception to make these differences seem large. The difference between the part of these phases which belong to one microcanonical ensemble of the whole system and the part which belongs to another would still be imperceptible, when the quan- tity of the bath is sufficiently increased. THERMODYNAMIC ANALOGIES. 183 As a general theorem, the conclusion may be expressed in the words : — If a system of a great number of degrees of freedom is microeanonieally distributed in phase, any very small part of it may be regarded as canonically distributed.* It would seem, therefore, that a canonical ensemble of phases is what best represents, with the precision necessary for exact mathematical reasoning, the notion of a body with a given temperature, if we conceive of the temperature as the state produced by such processes as we actually use in physics to produce a given temperature. Since the anomalies of the body increase with the quantity of the bath, we can only get rid of aU that is arbitrary in the ensemble of phases which is to represent the notion of a body of a given temperature by making the bath infinite, which brings us to the canonical distribution. A comparison of temperature and entropy with their ana- logues m statistical mechanics would be incomplete without a consideration of their differences with respect to units and zeros, and the numbers used for their numerical specification. If we apply the notions of statistical mechanics to such bodies as we usually consider in thermodynamics, for which the kinetic energy is of the same oider of magnitude as the unit of energy, but the number of degrees of freedom is enormous, the values of 0, de/dlogV, and d€/d wiU be of the same order of magnitude as 1/w, and the variable part of ^, log V, and will be of the same order of magnitude as n.f If these quantities, therefore, represent in any sense the notions of tem- perature and entropy, they wiU nevertheless not be measured in units of the usual order of magnitude, — a fact which must be borne in mind in determining what magnitudes may be regarded as insensible to human observation. Now nothing prevents our supposing energy and time in our statistical formulae to be measured in such imits as may * It is assumed — and without this assumption the theorem would hare no distinct meaning — that the part of the ensemble considered may be regarded as haying separate energy. t See equations (124), (288), (289), and (314) ; also page 106. 184 THERMODYNAMIC ANALOGIES. be convenient for physical purposes. But when these units have been chosen, the numerical values of O, de/dlo^Vy d€/d^ rjy log Vj j are entirely determined,* and in order to compare them with temperature and entropy, the nimierical values of which depend upon an arbitrary unit, we must mul- tiply all values of 6, de/dlogVy dejd^ by a constant (-ff), and divide all values of ^, log V^ and ^ by the same constant. This constant is the same for all bodies, and depends only on the imits of temperature and energy which we employ. For ordinary units it is of the same order of magnitude as the numbers of atoms in ordinary bodies. We are not able to determine the numerical value of JST, as it depends on the number of molecules in the bodies with which we experiment. To fix our ideas, however, we may seek an expression for this value, based upon very probable assimiptions, which will show how we would naturally pro- ceed to its evaluation, if our powers of observation were fine enough to take cognizance of individual molecules. If the unit of mass of a monatomic gas contains v atoms, and it may be treated as a system of 81; degrees of free- dom, which seems to be the case, we have for canonical distribution If we write T for temperature, and e^ for the specific heat of the gas for constant volume (or rather the limit toward which this specific heat tends, as rarefaction is indefinitely increased), we have t=c^. (492) since we may regard the energy as entirely kinetic. We may set the €p of this equation equal to the e^ of the preceding, * The unit of time only affects the last three quantities, and these only by an additive constant, which disappears (with the additive constant of entropy), when differences of entropy are compared with their statistical analogues. See page 19. THERMODYNAMIC ANALOGIES. 185 where indeed the individual values of which the average is taken would appear to human observation as identicaL This gives d® _ 2e^ dT" 3v' whence -^ = -g-^ • (493) a value recognized by physicists as a constant independent of the kind of monatomic gas considered. We may also express the value of JTin a somewhat different form, which corresponds to the indirect method by which physicists are accustomed to determine the quantity c^. The kinetic energy due to the motions of the centers of mass of the molecules of a mass of gas sufficiently expanded is easily shown to be equal to where p and v denote the pressure and volume. The average value of the same energy in a canonical ensemble of such a mass of gas is where v denotes the number of molecules in the gas. Equat- ing these values, we have 2?t; = 0v, (494) 1 pv whence -^ = -^—^^ (495) Now the laws of Boyle, Charles, and Avogadro may be ex- pressed by the equation pv=^AvTj (496) where J. is a constant depending only on the units in which energy and temperature are measured. 1 / JT, therefore, might be called the constant of the law of Boyle, Charles, and Avogadro as expressed with reference to the true number of molecules in a gaseous body. Since such numbers are imknown to us, it is more conven- ient to express the law with reference to relative values. If we denote by M the so-called molecular weight of a gas, that 188 SYSTEMS COMPOSED OF MOLECULES. the indentification of any particular particle of the first system with any particular particle of the second. And this would be true, if the ensemble of systems had a simultaneous objective existence. But it hardly applies to the creations of the imagination. In the cases which we have been con- sidering, and in those which we shall consider, it is not only possible to conceive of the motion of an ensemble of similar systems simply as possible cases of the motion of a single system, but it is actually in large measure for the sake of representing more clearly the possible cases of the motion of a single system that we use the conception of an ensemble of systems. The perfect similarity of several particles of a system will not in the least interfere with the identification of a particular particle in one case with a particular particle in another. The question is one to be decided in accordance with the requirements of practical convenience in the discus- sion of the problems with which we are engaged. Our present purpose will often require us to use the terms phase, densitf/^n-phasej statistical equUihrium, and other con- nected terms on the supposition that phases are not altered by the exchange of places between similar particles. Some of the most important questions with which we are concerned have reference to phases thus defined. We shall call them phases determined by generic definitions, or briefly, generic phases. But we shall also be obliged to discuss phases de- fined by the narrower definition (so that exchange of position between similar particles is regarded as changing the phase), which wiU be called phases determined by specific definitions, or briefly, specific phases. For the analytical description of a specific phase is more simple than that of a generic phase. And it is a more simple matter to make a multiple integral extend over all possible specific phases than to make one extend without repetition over aU possible generic phases. It is evident that if i^i, i/j . . . Vj^ are the numbers of the dif- ferent kinds of molecules in any system, the number of specific phases embraced in one generic phase is represented by the continued product [fi [j^ • • • [^> ^nd the coefficient of probabil- SYSTEMS COMPOSED OF MOLECULES. 189 ity of a generic phase is the sum of the probability-coefficients of the specific phases which it represents. When these are equal among themselves, the probability-coefficient of the gen- eric phase is equal to that of the specific phase multiplied by [i^ 1 1/, • • • \v^ It is also evident that statistical equilibrium may subsist with respect to generic phases without statistical equilibrium with tespect to specific phases, but not vice versa. Similar questions arise where one particle is capable of several equivalent positions. Does the change from one of these positions to another change the phase? It would be most natural and logical to make it afEect the specific phase, but not the generic. The number of specific phases contained in a generic phase would then be ]uj^ k^^ • • • hk ^^a"*? where #Cj, . . . Kj^ denote the numbers of equivalent positions belong- ing to the several kinds of particles. The case in which a a: is infinite would then require especial attention. It does not appear that the resulting complications in the formulae would be compensated by any real advantage. The reason of this is that in problems of real interest equivalent positions of a particle will always be equally probable. In this respect, equivalent positions of the same particle are entirely unlike the [i^^different ways in which v particles may be distributed in V different positions. Let it therefore be understood that in spite of the physical equivalence of different positions of the same particle they are to be considered as constituting a difference of generic phase as well as of specific. The number of specific phases contained in a generic phase is therefore always given by the product [f^lfi • • • [fV Instead of considering, as in the preceding chapters, en- sembles of systems differing only in phase, we shall now suppose that the systems constituting an ensemble are com- posed of particles of various kinds, and that they differ not only in phase but also in the numbers of these particles which they contain. The external coordinates of all the systems in the ensemble are supposed, as heretofore, to have the same value, and when they vary, to vary together. For distinction, we may call such an ensemble a grand erisembley and one in 190 SYSTEMS COMPOSED OF MOLECULES. which the systems differ only in phase a petit ensemble. A grand ensemble is therefore composed of a multitude of petit ensembles. The ensembles which we have hitherto discussed are petit ensembles. Let v^j . • . vj^j etc., denote the numbers of the different kinds of particles in a system, e its energy, and qij • • • qn^ i'l 9 • • • J'fi ^ts coordinates and momenta. If the particles are of the nature of material points, the nmnber of coordinates (n) of the system will be equal to 8 1'l . . . + Si';^. But if the parti- cles are less simple in their nature, if they are to be treated as rigid solids, the orientation of which must be regarded, or if they consist each of several atoms, so as to have more than three degrees of freedom, the number of coordinates of the system wiU be equal to the smn of ^i^v^^ etc., multiplied each by the nmnber of degrees of freedom of the kind of particle to which it relates. Let us consider an ensemble in which the number of systems having v^^ . . .vj^ particles of the several kinds, and having values of their codrdmates and momenta lying between the limits jj and q^ + dq^y p^ and p^ + dp^y etc., is represented by the expression Ne ® 1^1^ dp,...dq,, (498) where N, fl, 0, /tj , . . • /Lt^ are constants, N denoting the total number of systems in the ensemble. The expression Ne ^ (499) [vi...[v5 evidently represents the density-in-phase of the ensemble within the limits described, that is, for a phase specifically defined. The expression e ^ (500) SYSTEMS COMPOSED OF MOLECULES. 191 is therefore the probability-coefficient for a phase specifically defined. This has evidently the same value for all the Ifi.- • • [^ phases obtained by interchanging the phases of particles of the same kind. The probability-coefficient for a generic phase will be [j^, • . [i^ times as great, viz., e ® . (501) We shall say that such an ensemble as has been described is canonicallt/ distributed^ and shall call the constant & its modulus. It is evidently what we have called a grand ensem- ble. The petit ensembles of which it is composed are canonicaUy distributed, according to the definitions of Chapter rV, since the expression e ^ is constant for each petit ensemble. The grand ensemble, therefore, is in statistical equilibrium with respect to specific phases. If an ensemble, whether grand or petit, is identical so far as generic phases are concerned with one canonically distrib- uted, we shall say that its distribution is canonical with respect to generic phases. Such an ensemble is evidently in statistical equilibrium with respect to generic phases, although it may not be so with respect to specific phases. If we write H for the index of probability of a generic phase in a grand ensemble, we have for the case of canonical distribution H = ^ + f^in..^+,^HVH-e ^ ^g^3^ It wiU be observed that the H is a linear function of e and Vj, . . . 1/^ ; also that whenever the index of probability of generic phases in a grand ensemble is a linear function of €, i^i, . . . r^, the ensemble is canonically distributed with respect to generic phases. 192 SYSTEMS COMPOSED OF MOLECULES. The constant il we may regard as determined by the equation N=z l,...ly. / • • • / -^n i dpx... dq^, (604) or Ml»l"A«*y> ^ « ® = 2^1 . . . 2r. ^ 1— /•••/« ®^i>l • . • ^^fi'ml (^5) where the multiple sum indicated by 2^^ • • • S,^ includes all terms obtained by giving to each of the symbols vi . . . v^'dH integral values from zero upward, and the multiple integral (which is to be evaluated separately for each term of the multiple sum) is to be extended over all the (specific) phases of the system having the specified numbers of particles of the various kinds. The multiple integral in the last equation is _♦ what we have represented by e ®. See equation (92). We may therefore write _5 e e e = 2.,...2.,?^- r— • (506) It should be observed that the summation includes a term in which all the symbols v^. . . vj^ have the value zero. We must therefore recognize in a certain sense a system consisting of no particles, which, although a barren subject of study in itself, cannot well be excluded as a particular case of a system of a variable nimiber of particles. In this case € is constant, and there are no integrations to be performed. We have therefore* * Thii conclusion may appear a little strained. The original definition of ^ may not be regarded as fairly applying to systems of no degrees of freedom. We may therefore prefer to regard these equations as defining ^ in this case. SYSTEMS COMPOSED OF MOLECULES. 193 The value of e^ is of course zero in this case. But the value of 6g contains an arbitraiy constant, which is generally determined by considerations of conyenience, so that e^ and e do not necessarily vanish with v^,...v^. Unless — il has a finite Yalue, our formulae become illusory. We have already, in considering petit ensembles canonicaUy distributed, found it necessary to exclude cases in which — '^ has not a finite value.* The same exclusion would here make — ^jr finite for any finite values of vj . . . vj^. This does not necessarily make a multiple series of the form (506) finite. We may observe, however, that if for all values oi vi . . . vx — ^ ^ Co + e^ viy . . . + Cf^Vf,, (607) where Cq, c^y . . » c^ej^ constants or functions of 8, ~e e ® n Cn ''^'^^^ '**"^*, e e 6 ** e ^ I. ^.9 6 ^ ^M I.. • • . <2i '1 lij. •••-"»» (vj, fl Co iH-hH i*jt+»A i-«-. -2^l + « * ••• + « * • (508) 0—0 The value of — XI will therefore be finite, when the condition (507) is satisfied. If therefore we assume that — 11 is finite, we do not appear to exclude any cases which are analogous to those of naturct The interest of the ensemble which has been described Ues in the fact that it may be in statistical equilbrium, both in * See Chapter IV, page 36. t If the external coordinate! determine a certain yolume within which the system is confined, the contrary of (507) would imply that we could obtain an infinite amount of work by crowding an infinite quantify of matter into a finite yolume. 18 f 194 SYSTEMS COMPOSED OF MOLECULES. respect to exchange of energy and exchange of particles, with other grand ensembles canonically distributed and having the same values of O and of the coefficients /a^, fi^, etc., when the circumstances are such that exchange of energy and of particles are possible, and when equilibrium would not sub- sist, were it not for equal values of these constants in the two ensembles. With respect to the exchange of energy, the case is exactly the same as that of the petit ensembles considered in Chapter rV, and needs no especial discussion. The question of ex- change of particles is to a certain extent analogous, and may be treated in a somewhat similar manner. Let us suppose that we have two grand ensembles canonically distributed with respect to specific phases, with the same value of the modulus and of the coefficients Mi • • • Ma , and let us consider the ensemble of all the systems obtained by combining each system of the first ensemble with each of the second. The probabiUty-coefficient of a generic phase in the first ensemble may be expressed by e ® (509) The probability-coefficient of a specific phase will then be expressed by \ , ! > ♦ (610) once each generic phase compriBes [v^ . . . [»% specific phases. In the second ensemble the probability-coefficients of the generic and specific phases wiU be e « (611) // J^.. - //_-// fi"+Mli'l ...+Ma^ and ^ i-^^ ^, (512) SYSTEMS COMPOSED OF MOLECULES. 195 The probability-coefficient of a generic phase in the third ensemble, which consists of systems obtained by regarding each system of the first ensemble combined with each of the second as forming a system, wiU be the product of the proba- bility-coefficients of the generic phases of the systems com- bined, and will therefore be represented by the formula e ® (613) where ft'" = XI' -h fl", 6"f = e' + e", vi'" = i// -I- ri", etc. It wiU be observed that ri'", etc., represent the numbers of particles of the various kinds in the third ensemble, and e'" its energy ; also that XI'" is a constant. The third ensemble is therefore canonicaUy distributed with respect to generic phases. If all the systems in the same generic phase in the third ensemble were equably distributed among the | i/i'" , . • [y^"' spe-* cific phases which are comprised in the generic phase, the prob- ability-coefficient of a specific phase would be k'-'/j^ .. .- '// I .. — /'' -/// a'"+^»r^■^+H''^ — (514) In feet, however, the probability-coefficient of any specific phase which occurs in the thiid ensemble is e which we get by multiplying the probability-coefficients of specific phases in the first and second ensembles. The differ- ence between the formulae (614) and (615) is due to the fact that the generic phases to which (513) relates include not only the specific phases occurring in the third ensemble and having the probability-coefficient (515), but also all the specific phases obtained from these by interchange of similar particles between two combined systems. Of these the proba- 196 SYSTEMS COMPOSED OF MOLECULES. bility-coefficient is evidently zero, as they do not occur in the ensemble. Now this third ensemble is in statistical equilibrium, with respect both to specific and generic phases, since the ensembles from which it is formed are so. This statistical equilibrium is not dependent on the equality of the modulus and the co-effi- cients fi^y ... flJ^ in the first and second ensembles. It depends only on the fact that the two original ensembles were separ- ately in statistical equilibriimi, and that there is no interaction between them, the combining of the two ensembles to form n, third being purely nominal, and involving no physical connec- tion. This independence of the systems, determined physically by forces which prevent particles from passing from one sys- tem to the other, or coming within range of each other's action, is represented mathematically by infinite values of the energy for particles in a space dividing the systems. Such a space may be called a diaphragm. If we now suppose that, when we combine the systems of the two original ensembles, the forces are so modified that the energy is nc longer infinite for particles in all the space form- ing the diaphragm, but is diminished in a part of this space, so that it is possible for particles to pass from one system to the other, this will involve a change in the function c"' which represents the energy of the combined systems, and the equation e'" = e' + e" will no longer hold. Now if the co- efficient of probability in the third ensemble were represented by (613) with this new function e'", we should have statistical equilibrium, with respect to generic phases, although not to specific. But this need involve only a trifling change in the distribution of the third ensemble,* a change represented by the addition of comparatively few systems in which the trans- ference of particles is taking place to the immense number * It wiU be observed that, so far as the distribution is concerned, very large and infinite values of c (for certain phases) amount to nearly the same thing, — one representing the total and the other the nearly total exclusion of the phases in question. An infinite change, therefore, in the value of c (for certain phases) may represent a vanishing change in the distribution. SYSTEMS COMPOSED OF MOLECULES. 197 obtained by combining the two original ensembles. The difference between the ensemble which would be in statistical eqmlibrimn, and that obtained by combining the two original ensembles may be diminished without limit, while it is still possible for particles to pass from one system to another. In this sense we may say that the ensemble formed by combining the two given ensembles may stUl be regarded as in a state of (approximate) statistical equilibrium with respect to generic phases, when it has been made possible for particles to pass between the systems combined, and when statistical equilibrium for specific phases has therefore entirely ceased to exist, and when the equilibrium for generic phases would also have entirely ceased to exist, if the given ensembles had not been canonically distributed, with respect to generic phases, with the same values of and /tj, . • . flf^. It is evident also that considerations of this kind will apply separately to the several kinds of particles. We may diminish the energy in the space forming the diaphragm for one kind of particle and not for another. This is the mathematical ex- pression for a " semipermeable" diaphragm. The condition necessary for statistical equilibrium where the diaphragm is permeable only to particles to which the suffix ( )j relates will be fulfilled wlien fi^ and 6 have the same values in tiie two ensembles, although the other coefficients /tjv ^tc., may be different. This important property of grand ensembles with canonical distribution will supply the motive for a more particular ex- amination of the nature of such ensembles, and especially of the comparative nimibers of systems in the several petit en- sembles which make up a grand ensemble, and of the average values in the grand ensemble of some of the most important quantities, and of the average squares of the deviations from these average values. The probability that a system taken at random from a grand ensemble canonically distributed will have exactly vij . . . vj^ particles of the various kinds is expressed by the multiple integral 198 SYSTEMS COMPOSED OF MOLECULES. f- T [...'b ''""'''' ^'''^ or g ^ . (517) This may be called the probability of the petit ensemble (i/i, • . . 1/;^). The sum of all such probabilities is evidently unity. That is, n+Mi^i • ' ' +/*>»*-♦ e » Srj . . . Srjk j \ = 1> (518) which agrees with (506). The average value in the grand ensemble of any quantity w, is given by the formula n+Mi»'i...+M*»'*— « •n g tt = 2„^ . . . 2y^ / . . . / j j efpi . . • dq^. (519) J J h^ • • • \^h If ti is a function of v^^ . • .v^ alone, i. e., if it has the same value in all systems of any same petit ensemble, the formula reduces to Q+Miyi"»+MAyA-i Again, if we write tl] grand and m] peat to distinguish averages in the grand and petit ensembles, we shall have «*]g»ad = Sk, • . . 2„^ Sjpem r r (621) In this chapter, in which we are treating of grand en- sembles, V, will always denote the average for a grand en- semble. In the preceding chapters, u has always denoted the average for a petit ensemble. SYSTEMS COMPOSED OF MOLECULES. 199 Equation (505), which we repeat in a slightly different form, viz., r® = 2vj . . . ly^ f. . . f— . dpi . . . ^•, (522) J ^ J Is-- -lis shows that XI is a function of & and /x^, • . • /Lt^; also of the external coordinates a^, aj, etc., which are involved implicitly in 6. If we differentiate the equation regarding all these quantities as variable, we have A e Mll'l...+M»F»-€ aU S .^S.,...2.,J ...J i^TTTK -^'•••^' phases ■"- Ml"!- •+/*»»'»-« aU phases + etc. aU phases — etc. (623) a If we multiply this equation by e^, and set as usual A^, J^, etc., for — de/da^j — de/da^j etc., we get in virtue of the law expressed by equation (519), (f O O -^ cf , - - -\ - 0" + ^"^^ ="" ©^ 0*in . . • + M^v*-^ + ^^i + ^^» + «*^5 (624) 200 SYSTEMS COMPOSED OF MOLECULES. that is, Since equation (608) gives Q + Mivi " - + Ai»v> — i _ e /Ko/j\ — ^» (^26) the preceding equation may be written dQ = H(£0 - S ndfii — S Ji ctoi. (627) Again, equation (626) gives dQ + SA«id?i + S n d/Ai — di = ©dH + Hd®. (628) Eliminating d£l from these equations, we get d€ = — 0drH + S/Aicivi — SJficfai. (629) If we set ^ = € + ©H, (630) d^' = de + cBt + H d0, (631) we have d^' = H d& + S/Aid^i — S^idoi. (632) The corresponding thermodynamic equations are de = Tdrj + S/iidmi — S-4i dai, (633) ^ = € — Ti;, (634) rf,^ = — lydT+Sftidwh — S-^idoi. (636) These are derived from the thermodynamic equations (114) and (117) by the addition of the terms necessary to take ac- count of variation in the quantities (w^, tWj? ©tc.) of the several substances of which a body is composed. The cor- respondence of the equations is most perfect when the com- ponent substances are measured in such units that 9n^, m^ etc., are proportional to the numbers of the different kinds of molecules or atoms. The quantities /i^, fij' ©^-9 ^ these thermodynamic equations may be defined as differential coeffi- cients by either of the equations in which they occur.* * Compare Transactions Connecticut Academy, Vol. 111, pages 116 ft SYSTEMS COMPOSED OF MOLECULES. 201 If we compare the statistical equations (529) and (582) with (114) and (112), which are given in Chapter IV, and discussed in Chapter XIV, as analogues of thermody- namic equations, we find considerable difference. Beside the terms corresponding to the additional terms in the thermo- dynamic equations of this chapter, and beside the fact that the averages are taken in a grand ensemble in one case and in a petit in the other, the analogues of entropy, H and 17, are quite different in definition and value. We shall return to this point after we have determined the order of magnitude of the usual anomalies of v^, • • • Vj^. If we differentiate equation (518) with respect to /ai, and multiply by 0, we get ^— ^<£ + "Oh^TTq^ = '' (636) whence dCl/dfi^ = — y^, which agrees with (527). Differen- tiating again with respect to /i^, and to /^, and setting dii _ - dCl _ - dfii dfi^ we get The first members of these equations represent the average values of the quantities in the principal parentheses. ^We have therefore dKl ^ di (•-i - vi)* = n* - n = - 0^, = ^1 , (639) (n-n)(.'.-v.)=i7;;-nK.=-0^ = 0g = 0g. (640) 202 SYSTEMS COMPOSED OF MOLECULES. From equation (589) we may get an idea of the order of magnitude of the divergences of vi from its average value in the ensemble, when that average value is great. The equation may be written OlZl^'^®^^, (641) vi^ vi dfii The second member of this equation wUl in general be small when v^ is great. Large values are not necessarily excluded, but they must be confined within very small limits with re- spect to /Lu For if ^i^>^', (542) for all values of /x^ between the limits /l^' and /a^'^ we shall have between the same limits and therefore rjd?i><2Aiii (643) The difference /l^" — /aj' is therefore numerically a very small quantity. To form an idea of the importance of such a difference, we should observe that in formula (498) fii is multiplied by j/^ and the product subtracted from the energy. A veiy smaU difference in the value of /*i may therefore be im- portant. But since i/6 is always less than the kinetic energy of the system, our formula shows that /ij" — /Lt^', even when multiplied by vi' or i/i", may still be regarded as an insensible quantity. We can now perceive the leading characteristics with re- spect to properties sensible to human faculties of such an en- semble as we are considering (a grand ensemble canonically distributed), when the average numbers of particles of the vari- ous kinds are of the same order of magnitude as the number of molecules in the bodies which are the subject of physical SYSTEMS COMPOSED OF MOLECULES. 203 experiment. Although the ensemble contains systems having the widest possible variations in respect to the numbers of the particles which they contain, these variations are practi- cally contained within such narrow limits as to be insensible, except for particular values of the constants of the ensemble. This exception corresponds precisely to the case of nature, when certain thermodynamic quantities corresponding to 0, flu Ma' ^^"» which in general determine the separate densities of various components of a body, have certain values which make these densities indeterminate, in other words, when the conditions are such as determine coexistent phases of matter. Except in the case of these particular values, the grand en- semble would not differ to human faculties of perception from a petit ensemble, viz., any one of the petit ensembles which it contains in which vi^ Vg^ etc., do not sensibly differ from their average values. Let us now compare the quantities H and 17, the average values of which (in a grand and a petit ensemble respectively) we have seen to correspond to entropy. Since ^ _ Q + /xi yi . . . + /ij>n — g ^ ® and ^ = '^' H-iy = ^ + ^^"^";+^*"*""^- (646) A part of this difference is due to the &ct that H relates to generic phases and 17 to specific. If we write i/gen for the index of probability for generic phases in a petit ensemble, we have ^gen = ^ + log [11 . . . [vj , (646) H - 17 = H - i7,ea + log [n . . . til , (547) H — i7gen = g log [Vi . . . [VA . (548) This is the logarithm of the probability of the petit en- semble (I'l . . . Vj)J^ If we set « See formula (517). 204 SYSTEMS COMPOSED OF MOLECULES. 5^^' = Vr^, (549) which correspoDds to the equation 5f^' = ,. (660) we have ^^^ = ^ + ® log [i^ . . . [v*, and H-,^= " + ^^'''--'^+^*''*-V . (661) This -mR have a maximum when * Distinguishing values corresponding to this maximum by accents, we have approximately, when Vu • • • Vk ^^ ^^ ^i© same order of magnitude as the numbers of molecules in ordi- nary bodies, H-^.«. = ^ _ /^tA^Y(An)' /rfVeaVAnAva /rfV^Y(Av>)» V civi* y 20 Vfl?vifl?Fsy *" \ dvi,^ J 2© ' (553) '(554) where C = ""*" ^^'^' ' "^"*" ^*'*' " ^"°' , (555) and A vi = vi — vi', A vj = vj — vj', etc. (556) This is the probability of the system (I'l . . . vj^). The prob- abUty that the values of i/^ , . . . i/^^ lie within given limits is given by the multiple integral * Strictly speaking, ^^^ is not determined as function of i^i, . . . y^, except for integral values of these variables. Yet we may suppose it to be deter- mined as a continuous function by any suitable process of interpolation. SYSTEMS COMPOSED OF MOLECULES. 206 ... Te^e'Arf^i^ ) 26 WdJ e - U.a« ^ 2e ^^^...^,^. (567) This shows that the distribution of the grand ensemble with respect to the values of i^i, • . . vj^ follows the " law of errors " when y^', • ' -W *^re very great The value of this integral for the limits ± oo should be unity. This gives ? Y — — -L> 2>* 6' or h i C = ilog2>-Hlog(27r0), where 2> = \ ^y\ ) \dv\dv%) fiVs-Y f ^^-Y f^«~Y \dv%dv\) \ dv^ J • • • • \dv^dv^) that is, 2> = \ dv^ dv\ ) \dvj^ dv^ J \dvij \dyaj \dvi gen \dvij \dvtj (df^y \dyj \dyj (668) (559) (560) (561) f ^Y (-^y r^Y Now, by (668), we have for the first approximation H - ^^ = C = i log 2> - 1 log (27r©), (662) and if we divide by the constant jK^* to reduce these quanti- ties to the usual unit of entropy, H-^^ _ logD-h log (27r®) * See page 184-186. (563) 206 SYSTEMS COMPOSED OF MOLECULES. This is evidently a negligible quantiiy, since JTis of the same Older of magnitude as the number of molecules in oidinaiy bodies. It is to be observed that ^g^n is here the average in the grand ensemble, whereas the quantity which we vnah to compare with H is the average in a petit ensemble. But as we have seen that in the case considered the grand ensemble would appear to human observation as a petit ensemble, this dis- tinction may be neglected. The differences therefore, in the case considered, between the quantities which may be represented by the notations * are not sensible to human faculties. The difference and is therefore constant, so long as the numbers v-^y , , . vj^ are constant. For constant values of these numbers, therefore, it is immaterial whether we use the average of rj^^ or of rj for entropy, since this only affects the arbitrary constant of in- tegration which is added to entropy. But when the numbers Vj, . . . y^^ are varied, it is no longer possible to use the index for specific phases. For the principle that the entropy of any body has an arbitrary additive constant is subject to limi- tation, when different quantities of the same substance are concerned. In this case, the constant being detennined for one quantity of a substance, is thereby determined for all quantities of the same substance. To fix our ideas, let us suppose that we have two identical fluid masses in contiguous chambers. The entropy of the whole is equal to the sum of the entropies of the parts, and double that of one part. Suppose a valve is now opened, making a communication between the chambers. We do not regard this as making any change in the entropy, although the masses of gas or liquid diffuse into one another, and al- though the same process of diffusion would increase the *^In this paragraph, for greater distinctness, tigenL^^ and ilspecLtit ^*^® been written for the quantities which elsewhere are denoted bj H and ^. SYSTEMS COMPOSED OF MOLECULES. 207 entropy, if the masses of fluid were different. It is evident, therefore, that it is equilibrium with respect to generic phases, and not with respect to specific, with which we have to do in the evaluation of entropy, and therefore, that we must use the average of H or of i/gea, and not that of 17, as the equiva- lent of entropy, except in the thermodynamics of bodies in which the number of molecules of the various kinds is constant ^G^m^K: uBsu QC 175.04 C.1 cMnwnrafy pmo^Mw m tlMl ownora 3 6105 030 413 756 OAIEDUE STANFORD UNIVERSITY LIBRARIES STANFORD, CALIFORNIA 94305-6004 g