gale 'Bicentennial publication?
ELEMENTARY PRINCIPLES IN STATISTICAL MECHANICS
pale bicentennial publications
With the approval if tbt Prindent and FcUmn of Tali Unrveriity, a stria of volumes has keen prepared by a number of the Pnfesson and In- structorsj to be issued in connection with the Bicentennial Anniversary^ as a partial indica- ttm of the character of the studies in wbicb the University teachers are engaged.
This series of volumes is respectfully dedicated 4»
ff)r ^nurtures of tljr
ELEMENTARY PRINCIPLES
IN
STATISTICAL MECHANICS
DEVELOPED WITH ESPECIAL REFERENCE TO
THE RATIONAL FOUNDATION OF THERMODYNAMICS
BY
J. WILLARD GIBBS
Proftuor of Matktmatual Pkyrict in YaU University
OF r
UNIVERSITY
OF
NEW YORK : CHARLES SCRIBNER'S SONS
LONDON: EDWARD ARNOLD
1902
A<> '
Copyright, 1902, BY CHARLES SCRIBNER'S SONS
Published, March, zgoz.
UNIVERSITY PRESS • JOHN WILSON AND SON • CAMBRIDGE, U.S.A.
PREFACE.
THE usual point of view in the study of mechanics 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 continually taking place in the system. Inquiries of this kind are often simplified by taking into consideration conditions of the system other than 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.
94203
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 iiber das Verhalten mehratomiger Gasmolekiile mit Jacobi's Princip des letzten Multiplicators " (1871).
But although, as a matter of history, 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 quite 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 empirically 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 cur attention to the
X PREFACE.
phenomena distinctively 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 be 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 modulus 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 all 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 theoiy of the prop- erties of matter.
J. W. G.
NEW HAVEN, December, 1901.
CONTENTS.
CHAPTER I.
GENERAL NOTIONS. THE PRINCIPLE OF CONSERVATION
OF EXTENSION-IN-PHASE.
PAGE
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 II.
APPLICATION OF THE PRINCIPLE OF CONSERVATION OF 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 III.
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-PHASE CALLED CANONICAL, IN WHICH THE INDEX OF PROBABILITY IS A LINEAR
FUNCTION OF THE ENERGY.
PAGE
Condition of statistical equilibrium 32
Other conditions which the coefficient of probability must satisfy . 33
"""" Canonical distribution — Modulus of distribution 34
^ must be finite 35
The modulus of the canonical distribution has properties analogous
to temperature 35-37
Other distributions have similar properties 37
Distribution in which the index of probability is a linear function of
the energy and of the moments of momentum about three axes . 38, 39 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 39-41
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
= f 0 46, 47
Average value of total kinetic energy for any given configuration
or for the whole ensemble = % v 0 47
System of n degrees of freedom. Average value of kinetic energy,
for any given configuration or for the whole ensemble = f 0 . 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.
EXTENSION1-IN-CONFIGURATION AND EXTENSION-TN- VELOCITY.
Extension-in-configuration and extension-in-velocity are invari- ants . 57-59
CONTENTS. XV
PAGE
Dimensions of these quantities 60
Index and coefficient of probability of configuration 61
Index and coefficient of probability of velocity 62
Dimensions of these coefficients 63
Relation between extension-in-configuration and extension-in-velocity 64 Definitions of extension-in-phase, extension-in-configuration, and ex- tension-in- velocity, without explicit mention of coordinates . . 65-67
CHAPTER VII.
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 . . 80-83 General formulae relating to averages in a canonical ensemble . 83-86
CHAPTER VIII.
ON CERTAIN IMPORTANT FUNCTIONS OF THE ENERGIES OF A SYSTEM.
Definitions. V = extension-in-phase below a limiting energy (e).
$ = \o«dVldc 87,88
Vq = extension-in-configuration below a limiting value of the poten- tial energy (e?). fa = \o^dVqjdfq 89,90
Vp = extension-in-velocity below a limiting value of the kinetic energy
(*). ^p = loSdVpjd€p 90,91
Evaluation of Vp and $p 91-93
Average values of functions of the kinetic energy 94, 95
Calculation of FfromF^ 95,96
Approximate formulae for large values of n 97,98
Calculation of V or <£ for whole system when given for parts ... 98 Geometrical illustration . 99
xvi CONTENTS.
CHAPTER IX.
THE FUNCTION </> AND THE CANONICAL DISTRIBUTION.
When n > 2, the most probable value of the energy in a canonical ensemble is determined by d(j> j de = 1 / e 100,101
When n > 2, the average value of d$ j de in a canonical ensemble isl/e 101
When n is large, the value of <£ corresponding to d(f>/de=l/Q (<£o) js nearly equivalent (except for an additive constant) to the average index of probability taken negatively (— fj) . . 101-104
Approximate formulae for <£0 + fj 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 denned 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 123, 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 OF VARIOUS DIS- TRIBUTIONS IN PHASE.
Theorems I- VI. Minimum properties of certain distributions . 129-133 Theorem VII. The average index of the whole system compared with the sum of the average indices of the parts 133-135
CONTENTS. xvii
PAGE
Theorem VIII. 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 any limits 137-138
CHAPTER XII.
ON THE MOTION OF SYSTEMS AND ENSEMBLES OF SYS- TEMS THROUGH 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 XIII.
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 a priori foundation forthermo- dynamics requires mechanical definitions of temperature and entropy. Conditions which the quantities thus defined must satisfy 165-167
The modulus of a canonical ensemble (0), and the average index of probability taken negatively (rj), as analogues of temperature and entropy 167-169
xviii CONTENTS.
PAGE
The functions of the energy del d log Fand log Fas analogues of
temperature and entropy 169-172
The functions of the energy de / cty and <p as analogues of tempera- ture and entropy 1 72-1 78
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 183
Units of 0 and rj 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 (i>) . . . . 201
Average value of (v-v)* 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
(( UNIVERSITY J
ELEMENTARY PRINCIPLES IN STATISTICAL MECHANICS
CHAPTER I.
GENERAL NOTIONS. THE PRINCIPLE OF OF EXTENSION-IN-PHASE.
WE shall use Hamilton's form of the equations of motion for a system of n degrees of freedom, writing ql , . . ,qn for the (generalized) coordinates, qi , . . . qn for the (generalized) ve- locities, and
for the moment of the forces. We shall call the quantities Fl9...Fn the (generalized) forces, and the quantities p1 . . . pn, defined by the equations
Pl = ^-t p2 = ^, etc., (2)
dqi dq2
where ep 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 g's and ^'s. But in expressions like dep/dq1 , where the denominator does not determine the question, the kinetic
* The use of the momenta instead of the velocities 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.
4 HAMILTON'S EQUATIONS.
energy is always to be treated in the differentiation as function of the p's and q*s. We have then
* = ;fe* *l = -^ + Fl' etc> (3)
These equations will hold for any forces whatever. If the 'fetces^ &i*e £ dptt§erVative, in other words, if the expression (1) j.stant exact differential, we may set
where eq 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
e = €P + e«> (5)
and equatipns (3) may be written
*' = ;£' * = -£' etc- [I <«>
The potential energy (e3) may depend on other variables beside the coordinates q1 . . . qn. We shall often suppose it to depend in part on coordinates of external bodies, which we shall denote by ax , #2 , etc. We shall then have for the com- plete value of the differential of the potential energy *
deq = — FI dql . . — Fn dqn — A1 da^ — A2 daz — 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
de=qldpl . . . + qndpn~Pidqi . . .
— pn dqn — Al da-i — A2 daz — etc. (8)
It will be observed that the kinetic energy (e^,) in the most general case is a quadratic function of the p's (or g-'s)
* It will be observed, that although we call e 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. 5
v
involving also the ^'s but not the a's ; that the potential energy, when it exists, is function of the <?'s and a's ; and that the total energy, when it exists, is function of the jt?'s (or ^s), the 9's, and the a's. In expressions like dejdq^ them's, and not the q's, are to be taken as independent variables, as has already been stated with respect to the kinetic energy.
Lev 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 q19 . . . qn, either alone or with the coordinates a1? a2, etc. of certain external bodies. It is not necessary that they should be derivable from a force-function. The external coordinates a15 a2, etc. may vary with the time, but at any given time have fixed values. In this they differ from the internal coordinates q1 , . . . qn , 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 fall within specified limits of phase, viz., those for which
Pi <Pi< Pi", qi <qi< q",
Pn <Pn< P", qn < q» <
the accented letters denoting constants. We shall suppose the differences p^' — p{, q^ — q^, etc. to be infinitesimal, and that the systems are distributed in phase in some continuous manner,* so that the number having phases within the limits specified may be represented by
i') • • • (?»" - ?„'), (10)
* In strictness, a finite number of systems cannot be distributed contin- uously in phase. But by increasing indefinitely the number of systems, we may approximate to a continuous law of distribution, such as is here described. To avoid tedious circumlocution, language like the above may be allowed, although wanting in precision of expression, when the sense in which it is to be taken appears sufficiently clear.
6 VARIATION OF THE
or more briefly by
. . . dpn dql . . . dqn, (li)
where D is a function of the p's and q's and in general of t alb 3,
for as time goes on, and the individual systems change the\r
phases, the distribution of the ensemble in phase will in gen-
eral vary. In special cases, the distribution in phase will
remain unchanged. These are cases of statistical equilibr turn.
If we regard all possible phases as forming a sort oi exten-
ision of 2 n dimensions, we may regard the product of differ-
fentials in (11) as expressing an element of this extension, and
\D as expressing the density of the systems in that element.
We shall call the product
dpl... dpn dqlf. . dqn (12)
an element of extensionrin-phase, and D the density-inr-phase of the systems.
It is evident that the changes which take place in the den- sity of the systems in any given element of extension-in- phase will depend on" the dynamical nature of the systems and their distribution in phase at the time considered.
In the case of conservative systems, with which we shall be principally concerned, their dynamical nature is completely determined by the function which expresses the energy (e) in terms of the |?'s, <?'s, and a's (a function supposed identical for all the systems) ; in the more general case which we are considering, the dynamical nature of the systems is deter- mined by the functions which express the kinetic energy (ep) in terms of the p's and <?'s, and the forces in terms of the <?'s and «'s. The distribution in phase is expressed for the time considered by D as function of the p's and q's. To find the value of dD/dt for the specified element of extension-in- phase, we observe that the number of systems within the limits can only be varied by systems passing the limits, which may take place in 4 n different ways, viz., by the pl of a sys- tem passing the limit p^, or the limit p/', or by the ql of a system passing the limit q^ or the limit <?/', etc. Let us consider these cases separately.
DENSITY-IN-PHASE. 1
In the first place, let us consider the number of systems which in the time dt pass into or out of the specified element by pl passing the limit p^. It will be convenient, and it is evidently allowable, to suppose dt so small that the quantities ^ dt, ql dt, etc., which represent the increments of pl, ql, etc., in the time dt shall be infinitely small in comparison with the infinitesimal differences p£ — p^, q±r — <?/, etc., which de- termine the magnitude of the element of extension-in-phase. The systems for which pl passes the limit p^ in the interval dt are those for which at the commencement of this interval the value of p1 lies between p^ and p^ — p± dt, as is evident if we consider separately the cases in which pl is positive and negative. Those systems for which p1 lies between these limits, and the other p's and j's between the limits specified in (9), will therefore pass into or out of the element considered according aH^t? 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 of dt as a factor, and is evidently negligible, when dt
1 is sufficiently small, compared with the number which we are seeking to evaluate, and which (with neglect of terms contain- ing dt2) may be found by substituting pl dt for p^' — p^ in (10) or for dp1 in (11). The expression
Dpi dt dpz . . . dpn dqi . . . dqn (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 Pi), will 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 . . . dpn dqi . . . dqn 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 PI'.
The decrease in the number of systems within the limits due to systems passing the limits q± and <?/' may be found in the same way. This will give
for the decrease due to passing the four limits p±, p^", <?/, q^'. But since the equations of motion (3) give
^ + ^ = 0, (16)
dpi dql
the expression reduces to
(dD • dD • \ d^pi + d^ ?i) *• • • • *• dyi • • • *-•*• (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,
T~ i*
-dDdPl... dpn dSl... dqn, (18)
d~ ^ ) dpl ' ' ' d d ' " d dt ~
or
where the suffix applied to the differential coefficient indicates that the JP'S and <?'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 <?'s, (dD/dt}ptg vanishes, and therefore the condition will continue to hold, and the distribution in phase will be permanent, so long as the external coordinates remain constant. But the statistical equilibrium would in general be disturbed by a change in the values of the external coordinates, which
DENSITY-IN-PHASE. 9
would alter the values of tlie jt?'s as determined by equations (3), and thus disturb the relation expressed in the last equation. If we write equation (19) in the form
it will be seen to express a theorem of remarkable simplicity. Since D is a function of t, pl, . . . pn, ql , . . . qn, 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 ^'s), and the rest of the first member represents the increments of D due to increments of the p's and g's, expressed by pl dt, ql dt, etc. But these are precisely the increments which the jt?'s and #'s receive in the movement of a system in the tune 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 laws, but distributed in phase in any continuous manner, the density-in-phase 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
(fL.,=»-
where a, . . . h represent the arbitrary constants of the integral equations of motion, and are suffixed to the differential co-
* The condition that the forces Flt ...Fn are functions of q1 , . . . qn and alf a2, etc., which last are functions of the time, is analytically equivalent to the condition that Flf . . . Fn are functions of qi, ...qn and the time. Explicit mention of the external coordinates, a1? «2, etc., has been made in the preceding pages, because our purpose will require us hereafter to con- sider these coordinates and the connected forces, Alt A2, 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
JT.
.dpndqi... dqn (23)
taken within any limits the extension-in-phase within those limits.
When the phases bounding an extension-in-phase vary in the course of time according to the dynamical laws of a system subject to forces which are functions of the coordinates either alone or with the time, the value of the extension-in-phase thus bounded remains constant. In this form the principle may be called the principle of conservation of extension-in-phase. 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
/ . . . / D dp! . . . dp
If the extension is infinitely small, we may regard D as con- stant in the extension and write
D I . . . I dpl . . . dpn dq^ . . . dqn
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 . . . / fa . . . dpn dql . . . dqn,
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 coordinates 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 -Z>2'. 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 Zy. 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 will coincide with the third system of coordinates. This will give Dj' — ^Y'- 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 will 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
The analogue in statistical mechanics of this equation, viz.,
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% = Ds". We have therefore 2V = 2>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 coordinates which is used in its evaluation. This may easily be verified directly. If g1^ . . ,qn^ Qlt . . . Qn are two systems of coordinates, and Pi, • • • pn> P\i • - • Pn the cor- responding momenta, we have to prove that
J'...Jdp1...dpndqi...dqn=j*...fdPl...dPndQ1...dQn,(2£)
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
. . P., ft, . . . ft) = 1
>Pn>2i, •-• • 2V)
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(Ql9
We may transform any element of the first of these deter- minants as follows. By equations (2) and (3), and in view of the fact that the (j's are linear functions of the <?'s and therefore of the _p's, with coefficients involving the <?'s, so that a differential coefficient of the form dQr/dpy is function of the <?'s alone, we get *
* The form of the equation
d dep _ d dfp dpy dQx dQx dpv
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 py are not independent and that the proof depends on the linear relation between the Q's and the p's.
IS AN INVARIANT. 13
r dQx dpy
^^n/^dQL\=_d_de,==d^ dQx r^i W& %J d& cZft, d& '
But since f'0
r— i \ a (j/r /
d-k = ^. (28)
*& ^0.
Therefore,
...gn)
... Qn) The equation to be proved is thus reduced to
which is easily proved by the ordinary rule for the multiplica- tion of determinants.
The numerical value of an extension-in-phase will however depend on the units in which we measure energy and time. For a product of the form dp dq has the dimensions of energy multiplied by time, as appears from equation (2), by which the momenta are defined. Hence an extension-in-phase has the dimensions of the nth power of the product of energy and time. In other words, it has the dimensions of the nth power of action, as the term is used in the ' principle of Least Action.'
If we distinguish by accents the values of the momenta and coordinates which belong to a time ?, the unaccented letters relating to the time £, the principle of the conserva- tion of extension-in-phase may be written v * <•
//"» /* /%
... I dpi . . . dpndqi . . . dqn = I ... I dpj . . . dpnfdqir . . , dqn'} (31) *J *J *J
or more briefly
r r r
>!7 . . . dq^ (32)
14 CONSERVATION OF
the limiting phases being those which belong to the same systems at the times t and If respectively. But we have identically
/.../*,..., ,-/..
for such limits. The principle of conservation of extension-in- phase may therefore be expressed in the form
• • g«) -, xooN
..g.9 = 1'
This equation is easily proved directly. For we have identically
d(Pl,...qn) _ d(Pl,...qn)
• • • g.'O <*(M • • • g.O '
where the double accents distinguish the values of the momenta and coordinates for a time if'. If we vary t, while if and t" remain constant, we have
d_ d(Pl, ...qn) _ d(Pl"9 . . . qn") d_ d(Pl, ...qn)
Now since the time if' is entirely arbitrary, nothing prevents us from making if1 identical with t at the moment considered. Then the determinant
• • - ?»")
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. e., to the sum of the differentials of these elements. This gives the equation
d
_. dt d(pj>, . . . qn») dp," ' dpn" dqj* ' dqn»
Now since t = 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. 15
d d(plt ...
dtd(Pl»,...yn")
which substituted in (34) will give d
_ -
...n _
dtd(Pl',...qn')
The determinant in this equation is therefore a constant, the value of which may be determined at the instant when t = £', when it is evidently unity. Equation (33) is therefore demonstrated.
Again, if we write a, ... h for a system of 2 n arbitrary con- stants of the integral equations of motion, pv qv etc. will be functions of. a, ... h, and t, and we may express an extension- in-phase in the form
/rd(p "V «*(<
,, ^|T da - - • dh- (35>
d(a, ...h)
If we suppose the limits specified by values of 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 shall have a constant value. This requires that the determinant under the integral sign shall be constant, which may be written
...n dt d(a,...h) =°* (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
• • gj <*(pi, ...gn) d(pi', . . . qnr)
d(a, ...h) ' d(plf, . . . qn') d(a, ... h) in connection with equation (33).
Since the coordinates and momenta are functions of a, ... . h, and t, the determinant in (36) must be a function of the same variables, and since it does not vary with the time, it must be a function of a, ... h alone. We have therefore
„...*). ' (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 perfectly definite. If we write -ZVfor the total number of systems, and set
r = %. (38)
P may remain finite, when JV* and D become infinite. The integral
" * ... dqn (39)
taken within any given limits, will evidently 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
PdPl...dqn (40)
expresses the probability that an unspecified system of the ensemble will be found in the element of extension-in-phase dpi . . . dqn. 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 77. If we substitute NP and Ne1 for D in equation (19), we get
and
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
.,=°' (43)
(IX.... =°- (44)
That is, the values of P and rj, like those of D, 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 pi, • • • qn' at the time ^, and P" for that of the phase jp/', . . . qn" at the time tf',
2
18 CONSERVATION OF
J. . . JV dtf . . . dqj =f. . . Jp" dp{' . . . dqn", (45)
where the limits in the two cases are formed 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
P'f. . .fdPl> • • • <%»" =
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
P1' = Pf.
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- mined, 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 unity, it is evident that we must have
all
f...fpdPl...dqn = l, (46)
phases
where the integration extends over all phases. This is indeed only a different form of the equation
811
phases
which we may regard as defining
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 ct and the unit of energy is multiplied by ce , all indices of probability relating to systems of n degrees of
freedom will be increased by the addition of
•"-- n log ct + n log c€. (47)
CHAPTER II.
APPLICATION OF THE PRINCIPLE OF CONSERVATION
OF 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 applications 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 coordinates at the time t' fall between the limits pj and pj + dp^ q^ and q-L + dq^ etc., may be expressed by the formula
e* dPl' . . . dqj, (48)
where rf (the index of probability for the phase in question) is a function of the coordinates and momenta and of the time.
Let Qi, P^t etc. be the values of the coordinates and momenta which give the maximum value to ?/, and let the general value of rj be developed by Taylor's theorem according to ascending powers and products of the differences p^ — P/, Q.I ~ Ci'» Qte"> an(i let us suppose that we have a sufficient approximation without going beyond terms of the second degree in these differences. We may therefore set
n' = c — F', (49)
where c is independent of the differences p^ — P/, q{ — §/, etc., and F1 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
0=ef, (50)
we may write for the probability that the phase lies within the limits considered
dPl> . . . dqj. (51)
C is evidently the maximum value of the coefficient of proba- bility at the time considered.
In regard 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~F' is practically zero except for very small values of the differences Pi — P1/, q^ — Ci'> e^c< For very small values of these differences the approximation is evidently in general sufficient, for larger values of these differences the value of Ce~F' will be sensibly zero, as it should be, and in this sense the formula will represent the facts.
We shall suppose that the forces to which the system is subject are functions of the coordinates either alone or with the time. The principle of conservation of probability 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 t\ and that the phase (Pi', Qi'-) etc.) which has this greatest probability-coefficient, shall be that which corresponds to the phase (P/, §-/, etc.), i. e., 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 t" falls within the limits p^1 and p:" + dp^ #/' and #/' + cfy/', etc.,
" dpi" ...dqj', (52)
CONSERVATION OF+EXTENSION-IN-PHASE
where C represents the same value as in the preceding formula, viz., the constant value of the maximum coefficient of probability, and Fn is a quadratic function of the differences Pi ~ pi"> <?i" - Ci", etc., the phase (Px", QJ' etc.) being that which at the time t" corresponds to the phase (P/, #/, etc.) at the tune t'.
Now we have necessarily
J*. . .
&>i" . . . d£»" = 1, (53)
when the integration is extended over all possible phases. It will be allowable to set ± oo 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
l, (64)
Vf Vf"
where/' is the discriminant * of F1, and/" that of F". This discriminant is therefore constant in time, and like C an abso- lute invariant hi respect to the system of coordinates which may be employed. In dimensions, like (72, it is the reciprocal of the 2nth power of the product of energy and time.
Let us see precisely how the functions F' and F'f are related. The principle of the conservation of the probability-coefficient requires that any values of the coordinates and momenta at the time tf shall give the function F' the same value as the corre- _ sponding coordinates and momenta at the time tn give to F". Therefore Fn may be derived from F' by substituting for Pi* • • - 9.n their values in terms of p^', . . . <?/'. Now we have approximately
* This term is used to denote the 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 inF'.
AND THEORY OF ERRORS. 23
...+i^ (?."-<?."),
(55)
and as in IF" 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 (33) the eliminant of these equations has the value unity, the discriminant of F" will be equal to that of F1, 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 (33). At the time t\ the phases satisfying the equation
F' = k, (56)
where 7c is any positive constant, have the probability-coeffi- cient C e~k . At the time tf", the corresponding phases satisfy the equation
F" = k9 (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 C ' e~k, while phases without these limits have less probability-coefficients. The probability that the phase at the time tf falls within the limits F' — Jc is the same as the probability that it falls within the limits F" = k at the time t", 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 F = k by Z7, and the prob- ability that the phase falls within these limits by R, also the extension-in-phase within the limits F = 1 by Ur We have then by definition
F=k
l...dqn, (58)
24 CONSERVATION OF EXTENSION-IN-PHASE F—k
F=l
But since F is a homogeneous quadratic function of the differences
we have identically
F=k
rt
d(pi -Pi) . . . d(qn - Qn) kF=k
rwy&i
F=l
-Pj...d(<!.-Q1).
That is U=knUl} (61)
whence dU= U1nkn~ldk. (62)
But if k varies, equations (58) and (59) give
F=k-\-dk
dU = I . . . I dpi . . . dqn (63)
F=k
F=k+dk
F=k
Since the factor Oe~F has the constant value Ce~k in the last multiple integral, we have
dR = C e~kd U = C Ui n e~k kn~l dk, (65)
n e-k (\ + & + + . . . + N + const. (66)
We may determine the constant of integration by the condition that R vanishes with k. This gives
AND THEORY OF ERRORS. 25
(67)
R = C Z7i ]n - C U^ \n e~k fl + k + ~ + . . . + r^jY
We may determine the value of the constant U^ by the con- dition that R = 1 for k = oo. This gives (7 £7^ jw == 1, and
K = l _ e-k(l + A; + ^ . . . + [^ZTfV W
^«
(69)
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 k which substituted in equation (68) will give R = 1, the phases determined by the equation
F--=kB=i (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 III.
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 differential 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 shall not dis- tinguish between the coordinates and momenta. If we write rx . . . r2n for the coordinates and momenta, and a ... h 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
,...*). (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 (t) in the finite form, if we eliminate dt from these equations, we obtain 2^ — 1 equations in rl , . . . r2n and their differentials, the integration of which will introduce 2 n — 1 arbitrary constants which we shall call b ... h. If we can effect these integrations, the
* See Boltzmann: " Zusammenhang zwischen den Satzen iiber das Ver- halten mehratomiger Gasmoleciile mit Jacobi's Princip des letzten Multi- plicators. Sitzb. der Wiener Akad.,Bd. LXIII, Abth. II., S. 679, (1871).
THEORY OF INTEGRATION. 27
remaining constant (a) will then be introduced in the final integration, (viz., that of an equation containing dt,} 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 5, ... h and t — a are independent functions of rl , . . . r2n, the latter variables are functions of the former. The Jacobian in (71) is therefore function of 6, . . . ^, and t — a, and since it does not vary with t it cannot vary with #. We have therefore in the case considered, viz., where the forces are functions of the coordinates alone,
Now let us suppose that of the first 2 n — 1 integrations we have accomplished all but one, determining 2 n — 2 arbitrary constants (say c?, ... h) as functions of r^ , . . . r2n , leaving b as well as a to be determined. Our 2 w — 2 finite equations en- able us to regard all the variables r^ , . . . r2n, and all functions of these variables as functions of two of them, (say rl and r2,) with the arbitrary constants <?,... h. To determine 5, we have the following equations for constant values of <?, ... h.
u-/ 2 — ~~; — «** T ~77~ t*v*
da db
df^i , r2) c?7*2 7 dTi . .
whence -^7 — TT- db — — ^- dr^-\- -=— «r2. (74)
d(a, 6) c?a c?a
Now, by the ordinary formula for the change of variables,
= r
J
zn)
a^
28 CONSERVATION OF EXTENSION-IN-PHASE
where the limits of the multiple integrals are formed by the same phases. Hence
d(ri,rz) d(r^ ...rZn) d(c, ... h) d(a,b) " d(a,...h) d(r99...rj
With the aid of this equation, which is an identity, and (72), we may write equation (74) hi the form
The separation of the variables is now easy. The differen- tial equations of motion give rl and rz in terms of 'r^ , . . . r2n. The integral equations already obtained give <?,... h and therefore the Jacobian d(c, . . . A)/c?(r3, . . . r2n), in terms of the same variables. But in virtue of these same integral equations, we may regard functions of r19 . . . r2n as functions of rl and r% with the constants c, . . . h. If therefore we write the equation in the form
d(ri, . . .r2n) _ r2 ri ,
' ~ **- ..h) dr*> (77)
d(rs, • ..r2n) d(r8, . . . r2n)
the coefficients of drl and dr% may be regarded as known func- tions of rx and r2 with the constants <?,... h. The coefficient of db is by (73) a function of 6, . . . h. It is not indeed a known function of these quantities, but since <?,... h are regarded as constant in the equation, we know that the first member must represent the differential of some function of 5, ... A, for which we may write b'. We have thus
db' = r* dr ~ ..h) dr*> (78)
d(r8, . ..ran) d(r8, ...r2n)
which may be integrated by quadratures and gives V as func- tions of r1? r2 , ...<?,... A, and thus as function of r1? . . . r2n. This integration 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 form
Now, apart from any §uch considerations as have been ad- duced, if we limit ourselves to the changes which take place in time, we have identically
r2 dr± — r^ drz = 0,
and r± and r2 are given in terms of rv . . . r2n by the differential equations of motion. When we have obtained 2 n — 2 integral equations, we may regard r2 and r^ as known functions of rl and r2 . The only remaining difficulty is in integrating this equation. If the case is so simple as to present no difficulty, or if we have the skill or the good fortune to perceive that the multiplier
d(c,...h) ' (79)
d(r.,...rfc)
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 b' is a particular case of that represented by b. The system of arbi- trary constants «, 5', c . . . h has certain properties notable for simplicity. If we write b' for b in (77), and compare the result with (78), we get
= 1. (80)
d(a, b', c, . . . A) Therefore the multiple integral
da dbf do . . . dh (81)
30 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 rv . . . r2n, and t. We cannot use the princi- ple of conservation of extension-in-phase until we have made 2n — ~L integrations. Let us suppose that the constants 6, ... h have been determined by integration in terms of rv . . . r2w, and t, leaving a single constant (a) to be thus determined. Our 2 % — 1 finite equations enable us to regard all the variables rv . . . r2n as functions of a single one, say rr
For constant values of 5, ... A, we have
**-£* + ft* (82)
Now
* * \MI 1 , _
-5— da dr* . . . drzn =
t
da . . dh
d(a, ...h)
^"^ I • • • f "
J J d(a} ... A) d(r2, . . . r2n)
where the limits of the integrals are formed by the same phases. We have therefore
^'•••A>, (83)
da " d(a,...h) d(rt, . . . r,n) by which equation (82) may be reduced to the form
da =
M M a, . . . h) d(b, ... A)
d(r2, . . .
Now we know by (71) that the coefficient of da is a func- tion of a, ... h. Therefore, as £, ... h are regarded as constant in the equation, the first number represents the differential
AND THEORY OF INTEGRATION. 31
of a function of a, . . . h, which we may denote by a'. We have then
da'= d(b,...h) dr^~ d(b*..K) dt> (85)
dfa, ...r2n) d(r2, ...r2n)
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(rz, . . . rzn)
for the integration of the equation
dr, -rldt = 0. (87)
The system of arbitrary constants a', 5, ... h has evidently the same properties which were noticed in regard to the system a, 6', ... h.
CHAPTER IV.
ON THE DISTRIBUTION IN PHASE CALLED CANONICAL, IN WHICH THE INDEX OF 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 thermodynamics.
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 p's and q*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. (e)
will therefore satisfy the equation, as indeed appears identi- cally if we write it in the form
<Wd^_dP_de\ =0 dq1dpl dpldql)~
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 equations (20), (41), (42), also the paragraph following equation (20). The positions of any external bodies which can affect the systems are here supposed uniform for all the systems and constant in time.
J.
CANONICAL DISTRIBUTION. 33
cient 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.,
all
JP4>...- <*?» = !. (89)
phases
These considerations exclude
P = e X constant,
as well as
P = constant,
as cases to be considered.
The distribution represented by
(90)
or
where ® and i/r 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 ®, (a quantity of the same dimensions as e,) but the reverse, since it makes the distribution independent of the units employed. The negative sign of e is required by (89), which determines also the value of ^ for any given ©, viz.,
all f
~®
=f. . .f
e dp,... dqn . (92)
phases
When an ensemble of systems is distributed in phase in the manner described, i. e.^ when the index of probability is a
3
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 (®) 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
9 dpl . . . dqn (93)
taken within those limits. We may express the same thing by saying that the multiple integral expresses the probability 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 this integral into parts so small that the exponential factor may be regarded as constant in each. The value of ^r is therefore in- dependent of the system of coordinates employed.
It is evident that ty 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 -\Jr 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 e, so that, while in any given case the numerical values of ^r or e will be entirely indefinite until the zero of energy has also been fixed for the system considered, the difference ty — e will 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. 35
It is evident 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 will exclude certain cases, but not such apparently, as will affect the value of our results with respect to their bearing on ther- modynamics. It will 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 ®, 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 call 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 0 ,
* It will 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
which has the same modulus. Let qv . . .qm, pv . . . pm be the coordinates and momenta of A, and qm+l , . . . qm+n, pm+l , . . . pm+n those of £. Now we may regard the systems A and B as together forming a system 0, having m + n degrees of free- dom, and the coordinates and momenta q^ . . . <?,„+„, pv . . . pm+n. The probability that the phase of the system (7, as thus defined, will fall within the limits
dpi , . . . dpm+n, dq1 , . . . dqm+n
is evidently the product of the probabilities that the systems A and B will each fall within the specified limits, viz.,
(94)
We may therefore regard C as an undetermined system of an ensemble distributed with the probability-coefficient
(95)
an ensemble which might be defined as formed by combining each system of the first ensemble with each of the second. But since eA + €B is the energy of the whole system, and ^A and >/r B 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 C we add certain forces acting between A and .5, and having the force-function — eAB. The energy of the system C is now €A + €B + €ABI and an ensemble of such systems distributed with a density proportional to
(96)
would be in statistical equilibrium. Comparing this with the probability-coefficient of C given above (95), we see that if we suppose eAB (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 will differ infinitely little from one of statistical equilibrium, which is equivalent to saying that its distribution in 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 ®A and ®5. The prob- ability-coefficient of C would then be
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 will 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 rjr and 77" be the indices of probability in two independ- ent ensembles which are each in statistical equilibrium, then rf _j_ y wni De 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 vf + if1 will be a con- stant of motion. Now when infinitesimal forces are added to the compound systems, if r/ + rf1 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, V + ??" 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 will 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
• ^-I+|+S+S' (98)
where e denotes the energy of the system, and ©j , o>2 , &>3 , 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
where the letters have similar significations, and O, Ox , O2 , 113 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
k + ^-!±^ + SL±^ + 2d^ + a±3L-, (ioo)
vy i/j 1/2 »*a
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 infini- tesimal conservative forces of attraction or repulsion between particles in different shells, determined by the same law for all the systems, the functions o^ + &>', &>2 + o>2', and &>3 + w3' will remain constants of motion, and a function differing in- finitely little from el + 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 will be certain " normal " types of vibration 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 by 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. Phil. Trans., vol. XII, p. 547, (1878).
40 OTHER DISTRIBUTIONS
into parts relating separately to vibrations of these different types. These partial energies will be constants of motion, and if such a system is distributed according to an index which is any function of the partial energies, the ensemble will 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 coordinates, 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
^~i?'*'~if (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 will be represented by the index
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 n + m types of normal vibration, and n + m partial energies which are independent constants of motion. If all the original n + 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 will 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, will 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- tems, and the distribution represented will 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 indices.f
* It is interesting to compare the above relations with the laws respecting the exchange of energy between bodies by 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 limitations 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 (eq) depends not only upon the coordinates ql . . . qn which determine the configuration of the system, but also upon certain coordinates «i, «2, 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 — deqjdav — deqfda2, etc., while — deqjdqv ... — deq/dqn 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 qi , . . . qn , p\ , . . . pn , 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
although a similar equation would not hold for differentiations relative to the internal coordinates.
the periods are the same they must be distributed canonically with same modulus in order that the compound ensemble with additional forces may be in statistical equilibrium. * See especially Chapter I, p. 4.
OF AN ENSEMBLE OF SYSTEMS. 43
We always suppose these external coordinates to have the same values for all systems of any ensemble. In the case of a canonical distribution, i. e., 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
(105)
by which ty may be determined, the external coordinates, ax , 02, etc., contained implicitly in e, as well as ®,^are to be re- garded as constant in the integrations indicated. The equa- tion indicates that -fy 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 ^
/ v aii
f i ./. \ 1 /»
0
, \
(- I ^ + I «») = p
all
phases
all Jf
• • -/^ e~° dPi • • • dv- ~ ete-> (106)
phases t
or, multiplying by 0 e®, and setting
-^=^ -£=^ etc->
all
|d® = ^® f. . .f
ee
phases
—
i e ® dpl . . . dqn
phases
r r
i I . . .
phases
r * (• fcf
2J ...JA2e&dpl...dqn + etc. (107)
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
r M C fc! « =J • • • J u e & dPl... dqa. (108)
phases
Comparing this with the preceding equation, we have
<ty = £ d® - ~ d® - A! da^ - 22 da2 - etc. (109)
(jj) (y
Or, since fe— J = ,, (110)
and ^=^
d\f/ = yd® — AI da,i — >Z2 d«2 — etc. Moreover, since (111) gives
dty - c?e = ©cfy + ^©, (113)
we have also
dk — — ® drj — ^ ddi — A2 da2 — etc. (114)
This equation, if we neglect the sign of averages, is identi- cal in form with the thermodynamic equation
de + Alda1 + Az daz + etc. drj= —y— -, (115)
or
de = Td-rj — A! daL — Az da2 — etc., (H6)
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 (77) is a quantity which is
OF AN ENSEMBLE OF SYSTEMS. 45
only defined by the equation itself, and incompletely defined in that the equation only determines its differential, and the constant of integration is arbitrary. On the other hand, the 77 in the statistical equation has been completely defined as the average value in a canonical ensemble of systems of the logarithm of the coefficient of probability of phase.
We may also compare equation (112) with the thermody- namic equation
A^ = — T]dT—Aldal — Azda<i — etc., (117)
where ^r 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 further investigated the properties of an ensemble of systems distributed in phase according to the law which we are 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 dyi dzi . . . dxv dyv dzv,
and for the product of the differentials of the momenta ml dxi mi dyi m^ dz1 . . . mv dxv mv dyv mv dzv .
The product of these expressions, which represents an element of extension-in-phase, may be briefly written
mi dxi . . . mv dzv dxi . . . dzv ; and the integral
e @ mi dxi . . . mv dzv dxi . . . dzv (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
(119) and
e
0 =e & e 2€> •••« 20 . (120)
The potential energy (e3) is independent of the velocities, and if the limits of integration for the coordinates are inde- pendent of the velocities, and the limits of the several veloci- ties are independent of each other as well as of the coordinates,
VALUES IN A CANONICAL ENSEMBLE. 47
the multiple integral may be resolved into the product of integrals
C. . . C
mvdzv. (121)
This shows that the probability that the configuration lies within any given limits is independent of the velocities, and that the probability that any component velocity lies within any given limits is independent of the other component velocities and of the configuration. Since
* 2
f 4V«>, <& = vz^®, (122>
I/ 00
and
J
e 2 ® m* dx! = V^Ti-mx©8, (123>
the average value of the part of the kinetic energy due to the velocity x19 which is expressed by the quotient of these inte- grals, is J <H). 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 coordinates is 3 v or n. We have, therefore, for the average value of the kinetic energy of a system
ep = !„© = £ w©. (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 TT m ®)*,
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
C J
e 2& dXl. (125)
The expression becomes more simple when the velocity is expressed with reference to the energy involved. If we set
s=(^xl,
the probability that s lies between any given limits is expressed by
~S*ds. (126)
Here s is the ratio of the component velocity to that which would give the energy ® ; in other words, s2 is the quotient of the energy due to the component velocity divided by ®. 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 f (27r©)¥ f . . . /**.** . . . dzv (127)
for those limits, where M denotes the product of all 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 ep is a homogeneous quadratic function of the ^>'s, it may be divided into parts by the formula
_ 1 ^^p -I @£p /-I OQ\
ENSEMBLE OF SYSTEMS. 49
where e might be written for ep 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
/+» f+» de ^r .
• • • / i*l ~fo 6 dPl ' ' • dPn
_oo J —oo api
-=r- (129)
e ® dpi . . . dpn Now we have by integration by parts
ty-C
r °° PI <^~^- dPl = © r 4
,/ _oo api j _
By substitution of this value, the above quotient reduces to
— , which is therefore the average value of \P\— for the 2 dpi
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 dp1 . . . dqn for dp± . . . dpn in the multiple integrals.) This gives J n ® 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 points.
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 eq 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
50 AVERAGE VALUES IN A CANONICAL
for the moment of these forces, we have for the period of their action by equation (3)
* =-(^-d^ + Fl = - — + Fl
dqi dqi dqi
The work done by the force F± may be evaluated as follows :
r rd€ *
= I Pi dqt -f I y—dqit J J dq^
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 tune of the action of the forces is diminished.) We have therefore,
f* f* n f*
\ dqi = I pi £1 dt = I qi dpt=. — I Pi dpi . (131)
For since the p's are linear functions of the q's (with coeffi- cients involving the #'s) the supposed constancy of the <?'s and of the ratios of the <?'s will make the ratio fa/Pi constant. The last integral is evidently to be taken between the limits zero and the value of p1 in the phase originally considered, and the quantities before the integral sign may be taken as relating to that phase. We have therefore
i = ipl^Lt (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 Fl , . . . Fn 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 ep is a homogeneous quadratic function of the jo's, which is incapable 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. 51
linear functions of the JD'S.* The coefficients in these linear functions, like those in the quadratic function, must be regarded in the general case as functions of the <?'s. Let
2ep = <2 + w22... + iv2 (133)
where MJ . . . un are such linear functions of the p'a. If we write
for the Jacobian or determinant of the differential coefficients of the form dp/du, we may substitute
for dp1 . . . dpn
under the multiple integral sign in any of our formulae. It will be observed that this determinant is function of the <?'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 it'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 />'s, when the in- tegrations are to cover all values of the jt?'s (for constant #'s) once and only once, they must cover all values of the w's once and only once, and the limits will be ± oo for all the u'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 t*'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 w'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 limits of phase is thus reduced to the form
...<*«*«*&...%,. (134)
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
__ (135)
— --'
I/
e
00
and for the average of the whole kinetic energy, JTI©, as before.
The fractional part of the ensemble which lies within any given limits of configuration, is found by integrating (184) with respect to the w's from — oo to + oo . This gives
J f.
• da,
which shows that the value of the Jacobian is independent of the manner in which 2ep is divided into a sum of squares. We may verify this directly, and at the same tune obtain a more convenient expression for the Jacobian, as follows.
It will be observed that since the M'S are linear functions of the p's, and the jt?'s linear functions of the ^'s, the u's will be linear functions of the <?'s, so that a differential coefficient of the form du/dq will be independent of the q's, and function of the <?'s alone. Let us write dpxjduy for the general element of the Jacobian determinant. We have
ENSEMBLE OF SYSTEMS. 53
dpx d de d r=n de dur
duy duy dqx duy r—\ dur dqx
— r?" ( ^e dur\ d de _ duy
Therefore
d(p, ...pn) __d(u, .. . Q d(u, . . . u^) d(q, . . . qn)
and
^«. ^)
These determinants are all functions of the <?'s alone.* The last is evidently the Hessian or determinant formed of the second differential coefficients of the kinetic energy with re- spect to <?j , . . . qn. We shall denote it by Aj. The reciprocal determinant
which is the Hessian of the kinetic energy regarded as func- tion of the p's, we shall denote by Ap. If we set
e & = I . . . / e ® Ap dp,...dpn
+00 +00 —Mj2 . . . — «n2
f. . . C
e 20 dUl . . . dun = (27r©)§, (140)
and *, = * - fe (141)
* It will be observed that the proof of (137) depends on the linear relation
dur
between the u's and q's, which makes — — constant with respect to the differ-
dqx
entiations here considered. Compare note on p. 12.
54 AVERAGE VALUES IN A CANONICAL
the fractional part of the ensemble which lies within any given limits of configuration (136) may be written
• dql . . . dqn, (142)
where the constant tyq may be determined by the condition that the integral extended over all configurations has the value unity.*
* In the simple but important case in which Aj is independent of the ^'s, and €j a quadratic function of the q's, if we write ea for the least value of €q (or of e) consistent with the given values of the external coordinates, the equation determining \l/q may be written
— 00 00
If we denote by q±t . . . qn' the values of qi , . . . qn which give fq its least value ea , it is evident that eg — ea is a homogenous quadratic function of the differ- ences ?! — qi, etc., and that dqt, . . . 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_J
J. . .fe & dp! . . . dpn,
00 —CO
for which we have found the value Ap * (2 TT 9) 3. By the same method, or by analogy, we get
where A9 is the Hessian of the potential energy as function of the q's. It will be observed that A? 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 *g~ea
J ' ' -f (€Q — fa)e dql . . . dqn
—00 —00
+00 +eo * a
J . . .J e dqi . . . dqn
ENSEMBLE OF SYSTEMS. 55
When an ensemble of systems is distributed in configura- tion in the manner indicated in this formula, i. e., 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 distributed in configuration.
For any given configuration, the fractional part of the systems which lie within any given limits of velocity is represented by the quotient of the multiple integral
®dPl...dpn, or its equivalent
-
l--
taken within those limits divided by the value of the same integral for the limits ± oo. But the value, of the second multiple integral for the limits ± oo is evidently
We may therefore write
~~®~ du^ . . . dun, (143)
The evaluation of this expression is similar to that of
+00 +00 _!?
...sfe &dpl...dpn
+00 +00 _CJL f...fe &dPl...dpn
- 00 - CO
which expresses the average value of the kinetic energy, and for which we have found the value $ n 6. We have accordingly
«4-«a = 2na Adding the equation
*i> = 2ne> we have I — ea = n e.
\
\
56 AVERAGES IN A CANONICAL ENSEMBLE.
/„ ^p-fp •••je & **dPl...dpn, (144)
or again
r r^=^ i
I . . . / e < Ar^Ti • • • 4»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. e., 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
. dpndqi . . . dqn, (146)
may also be expressed in the form
. . dqndql . . . dqn. (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 entirely 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 determined entirely by the limiting configurations with the modulus. Now t|r, 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 typ by equation (140), and therefore, by (141), of tyg. 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
^ ...dgn (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 until they have presented themselves in the study of the leading problems.
58 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
(149)
or *» **&„. 1* (150)
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(Pl9 . . . P.) <%i . . . qn) d(ql9 ...qn)
..-) d(Ql9...Qn)
where ql , . . . q^ft , . . .pn and Ql , . . . Qn9 P1 , . . . Pn are two systems of coordinates and momenta.* It follows that
i>
= r
J
* See equation (29).
AND EXTENSION IN VELOCITY. 59
and
/Cfd(Ql, ... Qn)\% JT> Jp
' ' J \d(P^ ~^P}) ' *
"'<%>!... <jp.
= c.
J
>!,-.. W
The multiple integral •
>! . . . dpndqi . . . rf^, (151)
which may also be written
£1 . . . dqndqi . . . dqn, (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 extension-in- phase.* In like manner we may say that the multiple integral (148) expresses an extension-in-configuration, and that the multiple integrals (149) and (150) express an extensionrin- velocity. We have called
dpi . . . <Zp.<fyi . . . dqn, (153)
which is equivalent to
A-^! . . . dqndqt . . . dqn, (154)
an element of extension-in-phase. We may call
A^ ...dqn (155)
an element of extension-in-configuration, and
. . . dpn, (156)
See Chapter I, p. 10.
60 EXTENSION IN CONFIGURATION
or its equivalent
. . d, (157)
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-velocity. This is evident from the formulae (151) and (152) 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
Pi- ' -PnVl • --it
that is, as the nth power of energy, since every product of the form pl q1 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 nth power of time. Therefore an extension- in-configuration has the dimensions of the nth power of time multiplied 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 density-in-configuration. That is, if a certain con- figuration is specified by the coordinates q1 . . . qn, and the number of systems of which the coordinates fall between the limits q1 and ql + dql , . . . qn and qn + dqn is expressed by
D.A^Zi • • • *2n, (158)
AND EXTENSION IN VELOCITY. 61
Dq will be the density-in-configuration. And if we set
«*=ip (159)
where N denotes, as usual, the total number of systems in the ensemble, the probability that an unspecified system of the ensemble will fall within the given limits of configuration, is expressed by
e^dqt . . . dqn. (160)
We may call &* the coefficient of probability of the, configura- tion, and t]q 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.
. . . dgn. (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*q, it follows that the values of 7)q 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 in some simple cases, such as a system of material points, we cannot compare velocities in one configuration with velocities in another, and speak of their identity or difference except in a sense entirely artificial. We may indeed say that we call the velocities in one configuration the same as those in another when the quantities qlt ...qn 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 pi,...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 p1 and p1 + dpl , . . . pn and Pn + dpm the extension-in-velocity within those limits will be
. . . dpn, and we may express the probability in question by
e^\^dPl . . . dpn. (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
h . . . dpn, (163)
or its equivalent
J1. . .J*»*4Mb . . . dgn, (164)
taken within the given limits.
It follows that the probability that the system will fall within the limits of velocity, ^ and ^ + dq19 . . . qn and 2» + dq* is expressed by
e^^d^^.d^. (165)
The value of the integrals (163), (164) is independent of the system of coordinates and momenta which is used, as is also the value of the same integrals without the factor e1?; therefore the value of TJP must be independent of the system of coordinates and momenta. We may call e1? the coefficient of probability of velocity, and tjp the index of proba- bility of velocity.
AND EXTENSION IN VELOCITY. 63
Comparing (160) and (162) with (40), we get
eV* = P = el (166)
or rjq + IP = ^. (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 e1* and e1? have the dimensions of the reciprocals of extension-in-configuration and extension-in- velocity respectively, i. e., the dimensions of t~n e~* and e~», where t represent any tune, and e any energy. If, therefore, the unit of time is multiplied by ct, and the unit of energy by ce , every rjq will be increased by the addition of
n log ct + i?i log c. , (168)
and every rjp by the addition of
in logo.* (169)
It should be observed that the quantities which have been called extension-in-configuration and extension-in-velocity 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 measure 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 I.
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 St will be distributed throughout an extension in configuration equal to its original extension-in-velocity multiplied by $tn.
In demonstrating this theorem, we shall write q^ . . . qnf for the initial values of the coordinates. The final values will evidently be connected with the initial by the equations
Now the original extension-in-velocity is by definition repre- sented by the integral
J. . ,JV4i • - • <&, (171)
where the limits may be expressed by an equation of the form F(jll...^) = Q. (172)
The same integral multiplied by the constant St* may be written
J. . . jVd&ft), . . . %„&), (173)
and the limits may be written
(It will be observed that St as well as A^ is constant in the integrations.) Now this integral is identically equal to
f. . ./A,* d(q, - <?/) . . . d(q, . . . ft,'), (175)
or its equivalent
AM. • • • *» (176)
f. • -/
with limits expressed by the equation
/ (ft -<?/,••• 2.- 2,.') =0. (177)
AND EXTENSION IN VELOCITY. 65
But the systems which initially had velocities satisfying the equation (172) will after the interval Bt 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- velocity 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 coordinates. It will 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, Vl , . . . Vn of which a system in a given configuration is capable. We may conceive of the system as having a certain velocity F~0 combined with a part of each of these velocities Vl . . . Vn. By a part of V\ is meant a velocity of the same nature as V\ but in amount being anything between zero and Vr 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 V\ , . . . Vw 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 Fi , . . . Vn taken separately.
The more general case may be reduced to this simpler case as follows. The velocity F2 may always be regarded as composed of two velocities Vj and V2", of which VJ is of the same nature as Vl , (it may be more or less in amount, or opposite in sign,) while V2" satisfies the relation that the
5
66 EXTENSION IN CONFIGURATION
kinetic energy due to Vl and V2n combined is the sum of the kinetic energies due to these velocities taken separately. And the velocity VB may be regarded as compounded of three,
*Y» F3"> *Y"» of which v* is of the same nature as Fi ' V* of the same nature as V2", while VB"f satisfies the relations
that if combined either with Fi 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 Fg , . . . Vn have been thus decomposed, the square root of the product of the doubled kinetic energies of the several velocities PI> JY'» JY"» ete*' ^H 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 e12 for the kinetic energy of the velocities Fx 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 F2 . Let the mutual energy of every pair of the velocities Fj , . . . Vn be expressed in the same way. Analogy would make en represent the energy of twice V1 diminished by twice the energy of Fi , i. e.y en would represent twice the energy of Fi , although the term mutual energy is hardly appropriate to this case. At all events, let en have this signification, and e22 represent twice the energy of F^, etc. The square root of the determinant
n €12 ... €i
represents the value of the extension-in-velocity determined as above described by the velocities V\ , . . . FJ,.
The statements of the preceding paragraph may be readily proved from the expression (157) on page 60, viz.,
A •
by which the notion of an element of extension-in-velocity was
AND EXTENSION IN VELOCITY. 67
originally defined. Since A^ in this expression represents the determinant of which the general element is
the square of the preceding expression represents the determi- nant of which the general element is
Now we may regard the differentials of velocity dqt, d^ as themselves infinitesimal velocities. Then the last expression represents the mutual energy of these velocities, and
d*e
represents twice the energy due to the velocity dq{.
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 VII.
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
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
r au ^ *<r-*g =J...Jue " ^dqi...dqn} (179)
u
conflg.
where the integrations cover all possible configurations. The value of i|rg is evidently determined by the equation
r ^ r _!?
=J . . .J e %*dfc . . . dqn. (180)
e
config.
By differentiating the last equation we may obtain results analogous to those obtained in Chapter IV from the equation
£ - *" ~ f
0
J * J & &dPl ' ' '
e
«.
phases
As the process is identical, it is sufficient to give the results : dfa = rjqd® — J^i — J^da^ — etc., (181)
AVERAGES IN A CANONICAL ENSEMBLE. 69 or, since \f/q = 7g + ®^g, (182)
and <fyc = <£g 4- ^grf® + ®<fya, (183)
ckg = — ©cfyg — ^etai — J2^«2 — etc. (184)
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),
(185)
which gives ^ = Yp ~ *p ; (186)
we have also ^, = £ n ®, (187)
and by (140), fa = - \ n © log (2ir0). (188) From these equations we get by differentiation
<%=^d®, (189)
and <£, = — ® d^. (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 locis 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
* Pogg. Ann., Bd. CXVI, S. 73, (1862) ; ibid., Bd. CXXV, S. 353, (1865), See also Boltzmann, Sitzb. der Wiener.Akad., Bd. LXIII, S. 728, (1871). t Verwandlungswerth des Warmeinhaltes.
70 AVERAGE VALUES IN A CANONICAL
are also identical with those given by Clausius for the corre- sponding quantities.
Equations (112) and (181) show that if ty or ^rq is known as function of S and «x , a2 , etc., we can obtain by differentia- tion e or eq, and Aly AZy etc. as functions of the same varia- bles. We have in fact
* = *f-«i=:*f-e. (192)
The corresponding equation relating to kinetic energy,
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 ty or from tyq.
The average values of the squares or higher powers of the energies (total, potential, or kinetic) may easily be obtained by repeated differentiations of -\|r, ^, ^p1 or e, eg, e^, with respect to <t). By equation (108) we have
c = J . . .J « e <fe . . . dfc, (195)
phases
and differentiating with respect to ®,
phases
whence, again by (108),
de _ ? — \fe d®~~ ®2
ENSEMBLE OF SYSTEMS. 71
=
Combining this with (191),
In precisely the same way, from the equation
, ^...^n, (199)
(200)
config.
we may obtain
In the same way also, if we confine ourselves to a particular configuration, from the equation
/.all r ^ 1
= /•••/ €Pe Ap dpi . . . dpM (201)
J J
we obtain
€r J
veloc.
which by (187) reduces to
?=(!n»+Jn)®». (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
(.-•3" = ?_, (204)
identically. Accordingly
(205) In like manner,
(206)
Hence
G-l)2 = Gfl - I,)2 + (ep-ep)2. (208)
Equation (206) shows that the value of deg/d® can never be negative, and that the value of d2tyg/d®2 or drjq/d® 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 54, in which the potential energy is a quadratic function of the q's, and Ag independent of the <?'s, we should get for the potential energy
and for the total energy
We may also write in this case,
(fq — «a)2 n (e-e0)2~n'
ENSEMBLE OF SYSTEMS. 73
(209)
e-__ ?.
~?~ "»^~» + »5fp
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 deq/dep is of the same order of magnitude as n. Such values of deqjdep can only occur within intervals (ej1 — epf) which are of the or- der of magnitude of n~~\ unless it be in cases in which eg is in general of an order of magnitude higher than ep. Postponing for the moment the consideration of such cases, it will be in- teresting to examine more closely the case of large values of deq/dep within narrow limits. Let us suppose that for ej and epf the value of deq/dep is of the order of magnitude of unity, but between these values of "ep very great values of the differ- ential coefficient occur. Then in the ensemble having modulus @" and average energies ep" and es", values of eq sensibly greater than eqrl 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 eq as constant, but ® and therefore ^ as variable, we get
/drjq\ __1 dif/q \Itq — €q .
\d®)€-®~d® ©^~' whence by (192)
74 AVERAGE VALUES IN A CANONICAL
That is, a diminution of the modulus will diminish the 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 ep' and e^, values of eq sensibly less than eg' 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 O between ®' and ®", and of ep between ep' and ep/;, the individual values of eq will be practically limited to the interval between e«/ and eg'r.
In the cases which remain to be considered, viz., when deq/dep has very large values not confined to narrow limits, and consequently the differences of the mean potential ener- gies in ensembles of different moduli are in general very large compared with the 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 with the mean kinetic energy, will 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 deq/dep is not in general large.
It follows that to human experience and observation with respect to such an ensemble as we are considering, or with 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 order of magnitude as the number of molecules in the bodies subject to our observation and experiment, e — e, €P — £pi *q — % would be in 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 the ordinary 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 deq/d~ep takes a very large value. To human observation the effect would be, that in ensembles in which ® and ep had certain critical values, ~eq would be in- determinate within certain limits, viz., the values which would correspond to values of ® and ep 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.f
To obtain general formulae for the average values of powers of the energies, we may proceed as follows. If h is any posi- tive whole number, we have identically
phases phases
t. e., by (108),
_i ,, _i
(215)
Hence
and
* This implies that the kinetic and potential energies of individual systems 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 formally 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 h = 1, this gives
which agrees with (191). From (215) we have also
In like manner from the identical equation
all , «,
config. conflg.
(221)
--/ rf\^ -— we get i? = e 0 (^©2 ^ J e ® , (222)
and
With respect to the kinetic energy similar equations will hold for averages taken for any particular configuration, or for the whole ensemble. But since
the equation
reduces to
ENSEMBLE OF SYSTEMS. 77
We have therefore
<226> " <227>
*(228)
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 jt?'s and <?'s,
all f
phases
J. . . J[e(e _ i)» _ h (e _ ;)« ®2* J e~0dPl, ...dy.
(229) _ i_ x enyj
phases
i. e., by (108),
• (230)
* In the case discussed in the note on page 54 we may easily get
which, with eg — 60 — „ ®,
gives
rr^j» = Qe + e»^) («,-*j« = |Qe + « *
Hence c — eaft = c*.
Again (e - 60)» = e - ea + 02^ (e - ea)*-1,
which with e — e0 = n &
gives
(e - ««)* = (n 6 + 02^) (e - ea)*-1 = n (w 0 + 02^)*~J0,
hence {7^j» = ?^ + *> e».
78 AVERAGE VALUES IN A CANONICAL
or since by (218)
-e)»« = e(e-e)» - A <«-
In precisely the same way we may obtain for the potential energy
(63-i3)^ = @2^(e3- eq^ + h(eq- eq)^ ©2g. (232) By successive applications of (231) we obtain
(e - i)2 = (e-e)8 =•
(e - e)6 = J>5e + 15DeD*e + 10(D2€)2 + 15(Z)e)8 etc.
where D represents the operator ®'2d/d®. 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
d€p _ n
d®~2
the general formula reduces to
(ep - ep)™ = ©2 A (€p - ep)» + ±nh& (ep - ~ep)^ (233) or
(234)
ENSEMBLE OF SYSTEMS. 79
But since identically
the value of the corresponding expression for any index will be independent of <*) and the formula reduces to
we have therefore
etc.1
It will be observed that when i/r or e is given as function of O, all averages of the form e^ or (e — T)ft are thereby deter-
* In the case discussed in the preceding foot-notes we get easily
and
For the total energy we have in this case
l h ~
x±-Tx2 i Ve-J =n'
ft — €\ ° _ 2
etc.
rurxs iar A
• ou
-: .•
/
f.
J
* «»»
ENSEMBLE OP SYSTEMS.
The multiple integrals in
average rallies of the expressions In the brackets,
may therefore set equal to zero. The first gives
as already obtained. With this relation and (191) we get from the other equations
We may add for comparison equation (205), which might be derived from (236) by differentiating twice with respect to 8 :
The two last equations give
dl
(Al - Al)(e - e) = — (6 - €)'. (245)
e?e
If i/r or e is known as function of 0, Oj, Oj, etc*, (e — e)2 may be obtained by differentiation as function of the same variables. And if i|r, or Av or 17" is known as function of 8, O
(e — e) may be obtained by differentiation. But (^Al — A^y- and (^Al — A^) (^2 — A2) cannot be obtained in any
similar manner. We have seen that (e— e)2 is in general a vanishing quantity for very great values of TI, which we may regard as contained implicitly in 0 as a divisor. The same is
true of (A^ — A^) (e — e). It does not appear that we can
assert the same of (A-^ — -4X)2 or (Al — A^) (^2 — -42), since
6
82 AVERAGE VALUES IN A CANONICAL
a^ may be very great. The quantities dte/da^ an belong to the class called elasticities. The former expression represents an elasticity measured under the condition that while &J is varied the internal coordinates ql9 . . . qn all remain fixed. The latter is an elasticity measured under the condi- tion that when ax 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 Al 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 A^ for the average value of Al in systems of the ensemble which have any same energy, it will be determined by the equation
/ . . . J e ®
. . . dqn
where the limits of integration in both multiple integrals are two values of the energy which differ infinitely little, say e and
fc±
e + de. This will make the factor e & constant within the limits of integration, and it may be cancelled in the numera- tor and denominator, leaving
/•••/- -£-<&>! ...dqn
2H.= / / (247)
J...J*!...*.
where the integrals as before are to be taken between e and e + de. A^\f is therefore independent of ®, being a function of the energy and the external coordinates.
ENSEMBLE OF SYSTEMS. 83
Now we have identically
Al — Ai = (Ai — 2T)e) + (2T1 1 — -4)>
where Al — ~A^e 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 (Al — A^\f) (A^\ e — A^) 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
Atf. (248)
In the same way it may be shown that
(A, - Al) (e-e) = (^ - AJ (e - e). (249)
It is evident that in ensembles in which the anomalies of energy e — e may be regarded as insensible the same will be true of the quantities represented by A^\f — A^
The properties of quantities of the form A^\€ 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 results which have been given in this chapter.
Let u be any function of the internal and external coordi- nates with the momenta and modulus. We have by definition
**-.>,V:.fc!
u-J...Juee d^.^dq, (250)
phases
If we differentiate with respect to ®, we have du f a r/du u u e
d®=J J (35-3 <#--^i
phases
84 AVERAGE VALUES IN A CANONICAL
du _du uty-e) udif, d®~d® -- &— + ®d®'
Setting u = 1 in this equation, we get
d\f/ _ \i/ — € d®~ 0
and substituting this value, we have
du du ue ue
If we differentiate equation (250) with respect to a (which may represent any of the external coordinates), and write A
for the force — -^ , we get
__ ail t *.
du r r( du u dif/ u . \ 3-= /.../V-5- + ^^-+7v^) da J J \da © da 0 /
da
phases
du du or — = —
Setting w = 1 hi this equation, we get
Substituting this value, we have
du au uA uA
du du
or ®-r-®-r = ^2-uI=(u-u)(A-2). (255)
da aa
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
(€ + D) (u - u) = 0, (256)
where D represents the operator ®2 d/d®. Hence
(e + D)A (u - u) = 0, (257)
where h is any positive whole number. It will be observed, that since e is not function of ®, (e + D)h may be expanded by the binomial theorem. Or, we may write
(e + />) u = (e + D) u, (258)
whence (e + X>)* u = (e + D)h u. (259)
But the operator (e + D)*, although in some respects more simple than the operator without the average sign on the e, cannot be expanded by the binomial theorem, since e is a function of ® with the external coordinates. So from equation (254) we have
<26°)
whence (~ + J;)* (« - u) = 0 ; (261)
The binomial theorem cannot be applied to these operators.
Again, if we now distinguish, as usual, the several external coordinates by suffixes, we may apply successively to the expression u — u any or all of the operators
,
, etc. (264)
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± , A2 , etc., in all the operators.
If u is independent of the momenta, formulae similar to the preceding, but having eq in place of e, may be derived from equation (179).
CHAPTER VIII.
ON CERTAIN IMPORTANT FUNCTIONS OF THE ENERGIES OF A SYSTEM.
IN order to consider more particularly 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 J^the extension-in-phase below a certain limit of energy which we shall call e. That is, let
>x . . . dqn, (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
>i • • • dqn
taken without limits has a finite value,* the less value repre- sented by
e
/...
u
•
taken below a limiting value of 6, and with the e before the integral sign representing that limiting value, will also be finite. Therefore the value of V, which differs only by a constant factor, will also be finite, for finite e. It is a func- tion of e and the external coordinates, a continuous increasing
* This is a necessary condition of the canonical distribution. See Chapter IV, p. 35.
88 CERTAIN IMPORTANT FUNCTIONS
function of 6, which becomes infinite with e, and vanishes for the smallest possible value of e, or f or e = — oo, if the energy may be diminished without limit. Let us also set
dV <f> = log — • (266)
The extension in phase between any two limits of energy, ^ and e", will be represented by the integral
/ de. (267)
And in general, we may substitute e* de for dpl . . . dqn in a 2tt-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 *
* 0ffe, (268)
and that the average value in the ensemble of any quantity which only varies with the energy is given by the equation j
(269)
where we may regard the constant *fy as determined by the equation $
^» =l
6=00
— &
e de, (270)
F=0
In regard to the lower limit in these integrals, it will be ob- served that V= 0 is equivalent to the condition that the value of e is the least possible.
* Compare equation (93). t Compare equation (108).
J Compare equation (92).
OF THE ENERGIES OF A SYSTEM. 89
In like manner, let us denote by Vq the extension-in-configu- ration below a certain limit of potential energy which we may call eg. That is, let
• JV
(2T1)
the integration being extended (with constant values of the external coordinates) over all configurations for which the potential energy is less than eg. Vq will be a function of eq with the external coordinates, an increasing function of e3, which does not become infinite (in such cases as we shall con- sider *) for any finite value of eq. It vanishes for the least possible value of e?, or for eq = — oo , if eq can be diminished without limit. It is not always a continuous function of eg. In fact, if there is a finite extension-in-configuration of con- stant potential energy, the corresponding value of Vq will not include that extension-in-configuration, but if eq be in- creased infinitesimally, the corresponding value of Vq will be increased by that finite extension-in-configuration. Let us also set
(272)
The extension-in-configuration between any two limits of potential energy eq and eqf may be represented by the integral
(273)
whenever there is no discontinuity in the value of Vq as function of eq 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 hi general, with the restriction mentioned, we may substitute e^q deq for Aj dq1 . . . dqn in an w-fold 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
* If Vq were infinite^ for finite values of e,, V would evidently be infinite for finite values of e.
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 illusory on account of discontinuities in the values of Vq as function of eq by substituting for the given discontinuous function a continuous 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 Vq is, or may be regarded as, a continu- ous function of eq (within the limits considered), the proba- bility that the potential energy of an unspecified system lies between the limits eq and eq' is given by the integral
where ^ may be determined by the condition that the value of the integral is unity, when the limits include all possible values of eq. In the same case, the average value in the en- semble of any function of the potential energy is given by the equation
u = / ue d€q. (275)
Vq=0
When Vq is not a continuous function of eff, we may write d Vq for e*qdeg in these formulae.
In like manner also, for any given configuration, let us denote by Vp the extension-in-velocity below a certain limit of kinetic energy specified by ep. That is, let
V, = J.
(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 ep. Vp will evidently be a continuous increasing function of ep which vanishes and becomes infinite with e. Let us set
The extension-in-velocity between any two limits of kinetic energy ep and ep" may be represented by the integral
f
e*pdep. (278)
And in general, we may substitute e^p dep for A,* dpl . . . dpn or Ag* dql . . . dqn 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 under 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 ep. Since A^ is function of the coordinates alone, we have by definition
1...dpn (279)
the limits of the integral being given by ep. That is, if
ep = F(Pl,...Pa), (280)
the limits of the integral for ep = 1, are given by the equation F(Pl,...Pa) = \, (281)
and the limits of the integral for ep — a2, are given by the equation
=«'. (282)
But since F represents a quadratic function, this equation may be written
1 (283)
92 CERTAIN IMPORTANT FUNCTIONS
The value of Vp may also be put in the form
r, = ***f...f*&...*%. (284)
Now we may determine Vp for ep = 1 from (279) where the limits are expressed by (281), and FJ, for ep ,= a2 from (284) taking the limits from (283). The two integrals thus deter- mined are evidently identical, and we have
(285)
i. e., Vv varies as e/. We may therefore set
, n
Vp=Cep*> eP = n-Cep* j (286)
where C is a constant, at least for fixed values of the internal coordinates.
To determine this constant, let us consider the case of a canonical distribution, for which we have
_
where e& = (2-*®) 2.
Substituting this value, and that of e*' from (286), we get
(287)
Having thus determined the value of the constant (7, we may
OF THE ENERGIES OF A SYSTEM. - 93
substitute it in the general expressions (286), and obtain the following values, which are perfectly general :
~ *(289)
It will be observed that the values of Vp and <f>p for any given ep 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 coordinates 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, <&*, V, 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 e3 is a quadratic function of the q's, and Aj independent of the q'a. In this case we have
(2 ')*(«« -
P(Jn)
+ i)
94 CERTAIN IMPORTANT FUNCTIONS
falls within given limits is independent also of the value of the modulus, being determined entirely by 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 energy, either for the whole ensemble, or for any particular configura- tion, is given by
€p
—•£ ?-i ue 0e,2 dep *(291)
Thus:
^®"' if m + ^>°> t(292)
* The corresponding equation for the average value of any function of the potential energy, when this is a quadratic function of the ^'s, and A£ is independent of the q's, is
In the same case, the average value of any function of the (total) energy is given by the equation
Hence in this case
j .f m + n>0-
and = , if
ii f vy
If n = 1, e* = 2 ir and d^jde = 0 for any value of e. If n = 2, the case is the same with respect to 02.
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
-) /o _\9 ^2 ~j if w > 1 ; (294)
if n > 2 ; (295)
= ©. (296)
If n = 2, e*p = 2 TT, and d<j>p/dep = 0, for any value of ep. The definitions of F, V# and F^, give
(297)
where the integrations cover all phases for which the energy is less than the limit e, for which the value of Fis sought. This gives
V=CvpdVq, (298)
and ,-j-r €9=6
e* = -~ — f e^pdVn, (299}
de j
where Vp and e^p are connected with Vq by the equation
€p + eq = constant ~ e. (300)
If n > 2, e*? vanishes at the upper limit, i. e., for ep = 0, and we get by another differentiation
€q=€
We may also write
62= e
F= J "P;/9^, (302)
* r °=J
(303)
96 CERTAIN IMPORTANT FUNCTIONS
etc., when Vq is a continuous function of eq commencing with the value Vq = 0, or when we choose to attribute to Vq a fictitious continuity commencing with the value zero, as de- scribed on page 90.
If we substitute hi these equations the values of Vp and e^p which we have found, we get
^= r/il /^ <« - <«) <* ^« ' (304)
(305)
where e^« c?eg may be substituted for d Vq in the cases above described. If, therefore, n is known, and Vq as function of €p V and e^ 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 V= 0, even when this is not the case with respect to Vq and eq. The same is true of e^, when n > 2, or when n = 2 if Vq in- creases continuously with eq from the value Vq = 0.
The last equation may be derived from the preceding by differentiation with respect to e. Successive differentiations give, if h < } n + 1,
dhVjd<? is therefore positive if A < J n + 1. It is an in- creasing function of e, if h < Jw. If e is not capable of being diminished without limit, dhVjd^ vanishes for the least possible value of e, if h < \n. If n is even,
n
(307)
OF THE ENERGIES OF A SYSTEM. 97
That is, V is the same function of e# as - — - — — of e.
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
/ = e
* deq = 6**+** dep, (308)
where the variables are connected by the equation (300). The maximum value of <f)p + <f>Q is therefore characterized by the equation
(309)
de, de,
The values of ep and eq determined by this maximum we shall distinguish by accents, and mark the corresponding values of functions of ep and eq in the same way. Now we have by Taylor's theorem
If the approximation is sufficient without going beyond the quadratic terms, since by (300)
€P ~€P' = - (e* - «/)» we may write
+^(d^P\'(d\}'-\(^ii^
2 *.» (312>
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 eq — eqf may be regarded as a vanishing quantity. This gives
> _ A/+V /-ON
(313)
or
^V+^' + ilog(2,)-ilog[-(^)'-(^)']. (3U)
From this equation, with (289), (300) and (309), we .may determine the value of $ corresponding to any given value of e, when <j>q is known as function of eq.
Any two systems may be regarded as together forming a third system. If we have F or $ given as function of e for any two systems, we may express by quadratures J^and $ for the system formed by combining the two. If we distinguish by the suffixes ( )x, ( )2, ( )12 the quantities relating to the three systems, we have easily from the definitions of these quantities
=ff
(sis)
$12 | 04>*f7T7' / p^1 fj T7" / n^1 ' ^2x7 /O-1 £\
«/ «/ «y
where the double integral is to be taken within the limits
Vi = 0, V2 = 0, and el + e2 = e12 ,
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 e1 and e2 respectively.
It will be observed that these equations are identical in form with those by which F'and $ are derived from Vp or cf>p and Vq or <f>q, except that they do not admit in the general case those transformations which result from substituting for Vp or (f>p the particukr functions which these symbols always represent.
OF THE ENERGIES OF A SYSTEM. 99
Similar formulae may be used to derive Vq or <j>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
C
= I
01 02
e e
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 61 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 e2 as abscissas measured to the left, and determine a second curve by erecting the ordinates e^. We may suppose the distance be- tween the origins to be e12, the second origin being to the right if e12 is positive. We may determine a third curve by erecting at every point in the line (between the least values of ei and e2) 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^12. 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 e12 by moving one of these curves to the right or left. The third curve must be constructed anew for each different value of e12.
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 number having energies between the limits e and e + de. The expression
Ne
(317)
represents what may be called the density-in-energy. This vanishes for e = GO, for otherwise the necessary equation
(318)
could not be fulfilled. For the same reason the density-in- energy will vanish for e = — co, 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 n > 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 if n > 2, it must have a maxi- mum in such cases, in which the energy may be said to have
* See page 96.
THE FUNCTION 0. 101
its most common or most probable value, and which is determined by the equation
d(f> 1
de ©* ^ '
This value of d(f>/de is also, when n > 2, its average value in the ensemble. For we have identically, by integration by parts,
'''=!+4>r~
v'=o v=o
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 tfyan two degrees of freedom, the average value of d$/de in an eiis^ri^y canpni- / cally distributed is identical with the value of the same, ential coefficient as calculated for the most .eoavrooi'. < in the ensemble, both values being reciprocals of the modulus. Hitherto, in our consideration of the quantities F", V# Vp, <£, </V 4>pi we have regarded the external coordinates as constant. It is evident, however, from their definitions that V and <£ are in general functions of the external coordinates and the energy (e), that Vq and $g are in general functions of the external coordinates and the potential energy (eg). Vp and <f>p we have found to be functions of the kinetic energy (ep) alone. In the equation
-/
de, (322)
by which -vfr may be determined, O and the external coordinates (contained implicitly in <£) are constant in the integration. The equation shows that i|r is a function of these constants.
102 TH& FUNCTION <j> AND
If their values are varied, we shall have by differentiation, if n >2
v=o
+ dai f*4. e~e+*<le + da, f|* <f ®+V + etc. (323) ' J dci^ J da2
V=0 V=Q
(Since e* vanishes with F", when n > 2, there are no terms due to the variations of the limits.) Hence by (269)
or, since — ^ (325)
©
<fy = ^0 - © - dox - 0 da, - etc. (326)
ttCt^ tt^
Comparing iliis with (112), we get
The first of these equations might be written*
r) <328)
but must not be confounded with the equation
d+\ fdf\ (de\
^A,«~ W« W*..
which is derived immediately from the identity
=-\ L\ (330)
* 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 all the letters a1} «2, etc., or all except the one which is explicitly varied.
THE CANONICAL DISTRIBUTION. 103
Moreover, if we eliminate dty from (326) by the equation
d^ = 0^ + ^d® + de, (331)
obtained by differentiating (325), we get
de = -®dv-®!Jr-dal-®<Q-da2- etc., (332)
Cia-l OLa^,
or by (321),
. _^ = ^e + ^^ + ^^ + etc. (333)
de da, aa2
Except for the signs of average, the second member of this equation is the same as that of the identity
«ty = ^de + ?±dal + ^da2 + etc. (334)
de dal da2
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 e0. The same suffix will be applied to functions of the energy determined for this value. Our identity then becomes
= de0 + da, + da, + etc. (335)
\de J0 \dajo \da2J0
It has been shown that
?=(^=l, (336)
de \de)0 ©'
when n > 2. Moreover, since the external coordinates have constant values throughout the ensemble, the values of d(p/dav d(f>Jda^ etc. vary in the ensemble only on account of the variations of the energy (e), 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
<25 ~d4
-5-S -=-S etc.,
104 THE FUNCTION <£ AND
as practically equivalent to the values relating to the most common energy
— — I j ( — j j etc. dtti JQ \ d&z J Q
In this case also de is practically equivalent to deQ. We have therefore, for very large values of n,
— dri — d<f>Q (337)
approximately. That is, except for an additive constant, — 77 may be regarded as practically equivalent to <£0, when the number of degrees of freedom of the system is very great. It is not meant by this that the variable part of rj + <£0 is numerically of a lower order of magnitude than unity. For when n is very great, — 77 and $0 are very great, and we can only conclude that the variable part of 77 + <£0 is insignifi- cant compared with the variable part of rj or of <£0, taken separately.
Now we have already noticed a certain correspondence between the quantities ® and 77 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 <f> and de/dQ 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 the function <j> in a series arranged according to ascending powers of e — e0. This ex- pansion may be written
(f £)
(€ ~ ^
(338)
Adding the identical equation
THE CANONICAL DISTRIBUTION. 105
\/ — 6 ^ — €Q 6 —
© © © >
(339)
Substituting this value in
which expresses the probability that the energy of an unspeci- fied system of the ensemble lies between the limits e' and e", we get
-0
**. (340)
When the number of degrees of freedom is very great, and e — e0 in consequence very small, we may neglect the higher powers and write*
i .
" (341)
This shows that for a very great number of degrees of freedom the probability of deviations of energy from the most probable value (e0) approaches the form expressed by the 'law of errors.' With this approximate law, we get
* If a higher degree of accuracy is desired than is afforded by this formula, it may be multiplied by the series obtained from
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 <j> AND
(343) whence
(344)
Now it has been proved in Chapter VII that
7 - ^ _2
(6 ~~ e) — ~ ~r~ €P ' n dep p
We have therefore
approximately. The order of magnitude of rj — <£0 is there- fore that of log n. This magnitude is mainly constant. The order of magnitude of rj + <pQ — \ log n is that of unity. The order of magnitude of </>0 , and therefore of — 77, is that of n.*
Equation (338) gives for the first approximation
(1^ = _£, (346)
(*-*>(.-0 = ^ = £*, W
/ . __ , Y — (6 ~ 6o)2 = ^ ^f (348)
a€p
The members of the last equation have the order of magnitude of n. Equation (338) gives also for the first approximation
de fi\ ~ \ ^2 / v€ eo)> * 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 (77). In any case, we must have
(351)
where e^+t is necessarily positive. This requires that it shall vanish for e = oo , and also for 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 n > 2, e* will vanish for that least value of e. In general therefore 77 + <£ will have a maximum, which determines the most probable value of the energy. If we denote this value by e0> and distinguish the corresponding values of the func- tions of the energy by the same suffix, we shall have
-a
The probability that an unspecified system of the ensemble
* We shall find hereafter that the equation
is exact for any value of n greater than 2, and that the equation
fd(f> IV __ <^0 \d* ®) ' rf? is exact for any value of n greater than 4.
108 THE FUNCTION <£ AND
falls within any given limits of energy (e' and e") is repre- sented by
f
e^de.
If we expand 77 and <£ in ascending powers of e — e0, without going beyond the squares, the probability that the energy falls within the given limits takes the form of the « law of errors ' —
de. (353)
i/
This gives
We shall have a close approximation in general when the quantities equated in (355) are very small, i. e., when
is very great. Now when n is very great, — d*$/de* 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 77.
We may obtain other properties pertaining to average values in a canonical ensemble by the method used for the average of d<j>/de. 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
6=00 4,-e
/- — - + 4> ue e de. (357)
F=0
THE CANONICAL DISTRIBUTION. 109
Now we have identically
Therefore, by the preceding equation
If we set u = 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
^ = i, (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 necessarily 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 canonically distributed, is
^ + M^4.M^- \uef¥*~\*=*> df U de U de ~ I"* J F=0
where t\ denotes, as usual, the index of probability of phase.
t The term finite applied to the modulus is intended to exclude the value zero as well as infinity.
110 THE FUNCTION 0 AND
The impossibility of a canonical distribution occurs when the equation
e e
e = e
s* — l-j-0
=J e ' de (361)
F=0
fails to determine a finite value for ^. Evidently the equation cannot make ty an infinite positive quantity, the impossibility therefore occurs when the equation makes ty = — oo . Now we get easily from (191)
If the canonical distribution is possible for any values of ®, we can apply this equation so long as the canonical distribu- tion is possible. The equation shows that as ® is increased (without becoming infinite) — ty cannot become infinite unless 6 simultaneously becomes infinite, and that as O is decreased (without becoming zero) — ^ 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 (361) has a finite value.
When this condition is fulfilled, the second member of (359) will vanish for u = e~+ V. For, if we set 6' = 26,
? ___! € _ f _ ^ ¥
F = 0 V = 0
THE CANONICAL DISTRIBUTION. Ill
where tyr denotes the value of ^ for the modulus ®'. Since the last member of this formula vanishes for e = oo , the less value represented by the first member must also vanish for the same value of e. 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 e, which gives V — 0, whether this be finite or negative infinity. Therefore, the second member of (359) vanishes at the lower limit also, and we have
V
or e V=®. (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 iT^ V and the notion of temperature in thermodynamics. We shall return to this subject in Chapter XIV.
If n > 2, the second member of (359) may easily be shown to vanish for any of the following values of u viz. : </>, e^, e, e"*, where m denotes any positive number. It will also vanish, when n > 4, for u = dfyde, and when n > 2 h for u = e-* dhV/d^. When the second member of (359) van- ishes, and n > 2, we may write
We thus obtain the following equations : If n > 2,
(364)
112
THE FUNCTION </> AND
or
If w > 4,
If n
®2
-<t>dhVd<f> 1 - 6 -d?-fc-®6
e ' -Tjr-j — e de1 ae
or
(368) t(369)
(370)
whence " ^- = ^.
Giving A the values 1, 2, 3, etc., we have
as already obtained. Also
* This equation may also be obtained from equations (252) and (321). Compare also equation (349) which was derived by an approximative method, t Compare equation (360), obtained by an approximative method.
THE CANONICAL DISTRIBUTION. 113
If Vq is a continuous increasing function of eg, commencing with Vq = 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 \jr have the suffix ( )ff. The equation may be transformed 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 valid- ity must be differently stated. The equation
requires only the condition already mentioned with respect to Vq. This equation corresponds to (362), which is subject to no restriction with respect to the value of n. We may ob- serve, however, that V will always satisfy a condition similar to that mentioned with respect to Vr
If Vq satisfies the condition mentioned, and e^ a similar condition, i. e., if e^i is a continuous increasing function of e3, commencing with the value (^ = 0, equations will hold sim- ilar to those given for the case when n > 2, viz., similar to (360), (364)-(368). Especially important is
deq ~®'
If Vq, 6*4 (or dVq/d€q), d?Vq/de* all satisfy similar conditions, we shall have an equation similar to (369), which was subject to the condition n > 4. And if cPVqjdef also satisfies a similar condition, we shall have an equation similar to (372), for which the condition was n > 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 n > 2 h.
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 F", similar to those given relating to the differential coefficients of Vq, instead of the conditions relating to n, for the validity of equations (360), (363)-(372). 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.
AN 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 limiting 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 density-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 obtained 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 dpl . . . dqn, 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 ensemble distributed with a density proportional to
where &> and e1 are constants, and then diminish indefinitely the value of the constant &>. 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. 117
(eq) vary in the different systems, subject of course to the con- dition
€p -f eq = e = constant. (373)
Our first inquiries will relate to the division of energy into these two parts, and to the average values of functions of ep and eq.
We shall use the notation y\ 6 to denote an average value in a microcanonical ensemble of energy e. An average value in a canonical ensemble of modulus (D, which has hitherto been denoted by M, we shall in this chapter denote by '^@, to distinguish more clearly the two kinds of averages.
The extension-in-phase within any limits which can be given in terms of ep and eq 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
/» r
udVpdVq
dVpdVq
Since d Vp = e^p dep, and dep = de when eq is constant, the expression may be written
To get the average value of u in an ensemble distributed microcanonically with the energy 6, we must make the in- tegrations cover the extension-in-phase between the energies e and e + de. This gives
118 A PERMANENT DISTRIBUTION IN WHICH
de\ueVpdVq
vq=o
But by (299) the value of the integral in the denominator is e^. We have therefore
(374)
where e^p and Vq are connected by equation (373), and w, if given as function of ep, or of ep and eq, becomes in virtue of the same equation a function of eq alone.
We shall assume that e^ has a finite value. If n > 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 — ty infinite.* When n > 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 n = 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 case of the microcanonical distribution is afforded by 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~* V. (375)
But by equations (288) and (289)
•-V,-?*. (376)
Therefore
e~* V— e~ P "Pjj e = - ep\e . (377)
Again, with the aid of equation (301), we get
= £» (378)
Vq=0
if n > 2. Therefore, by (289)
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 (306), (374), (288) and (289). We have
• ' ' r : , . w <
f*
=J
The equation may be written
€g=<
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 value of the energy will be found in any specified part of the circle?
120 A PERMANENT DISTRIBUTION IN WHICH We have therefore
if h < J- n + 1. For example, when w is even, we may make A = i- n, which gives, with (307),
1-2
(381)
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 e is a constant value in a microcanonical ensemble, and therefore identical with its average value. We get thus the general equation
.-*£?
if h < J n + 1.* The equations
. 9 _
(383)
may be regarded as particular cases of the general equation. The last equation is subject to the condition that n > 2.
The last two equations give for a canonical ensemble, x if n > 2,
(l-|)^leV^]0-l. (385)
The corresponding equations for a microcanonical ensemble give, if n > 2,
\l 1 A 1 ' _1| ^V* /OQ£\
I1 - = I W« V> = ^wTF' (386)
See equation (292).
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 rekting 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
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 e1 and ea is given by the quotient
which may also be written f
If we make the limits of integration e and e + de, we get the
* If this condition is rigorously fulfilled, the parts will have 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 IV, pp. 35 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 all 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)
where fa and V2 are connected by the equation
€i + €2 = constant = e, (388)
and u, if given as function of ei , or of ei and e2 , becomes in
virtue of the same equation a function of e2 alone.*
Thus
Je = e+ J F! rf F2 , (389)
(390)
This requires a similar relation for canonical averages
© = e~+ V\e = e^rje = e~+*V\* . (391)
Again
e2=e
SB =e-*f ^V'rfF,. (392)
del |e J del
F^O
But if w: > 2, «*i vanishes for Fj = 0,f and
. (393)
de
Hence, if n^ > 2, and w2 > 2,
d<f> _ dfal _ dfa\ /qq ..
^e ~ ^ |f ~ dez \f
* In the applications of the equation (387), we cannot obtain all the results corresponding to those which we have obtained from equation (374), because <t>p is a known function of ep, while fa must be treated as an arbitrary func- tion of €j , or nearly so.
t See Chapter VIII, equations (306) and (316).
ALL SYSTEMS HAVE THE SAME ENERGY. 123
and s « 5l =^\ = ^ • (395)
© de |0 rfej J0 rfe2 |e
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, ep, and eg, and the kinetic energies of the parts by e^, and e2p. 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 eg, e^, ezp may be represented in the notations of Chapter VIII by the triple integral
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 eq, e^, e^ will be expressed by the quotient
fffue^ded VZpd Vq
or
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
C2p=C— Cq
(396)
124 A PERMANENT DISTRIBUTION IN WHICH where 0^, V2P, and Vq are connected by the equation
€ip + €2p + eq = constant = e. Accordingly
J VlpdV2p dVq = e-* V, (397)
and we may write
;r 2 , 2 j /onON
2p|6 = — e^l€ = -€^|e, (398)
and
O f)
r \ _ _ ^ — I __ ^ — | ('399')
Again, if wx > 2,
C9=€ (ft (ft
~* C^'jir "*"**
= e J ^dFi=« ir*
Hence, if ^ > 2, and w2 > 2,
_ 2p _ f i 1 N -1) _ /I w -IN f -11
de ~de~l '* "~ '' p ^€ ~ ^ ~~ ' p '€
We cannot apply the methods employed in the preceding pages to the microcanonical averages of the (generalized) forces Av Ay, 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
(e - c'V
where ef 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
all
c— •
e, w dpl . . . dqn — 1, (403)
phases
or e =...e dPl . . . dqn, (404)
phases
_c | g
or again e = C e ^ de. (405)
From (404) we have
all
phases
= 00
, j
^ (406)
where H7ie denotes the average value of A1 in those systems of the ensemble which have any same energy e. (This is the same thing as the average value of A l in a microcanoni- cal ensemble of energy e.) The validity of the transformation is evident, if we consider separately the part of each integral which lies between two infimtesimally differing limits of energy. Integrating by parts, we get
126 A PERMANENT DISTRIBUTION IN WHICH
Jr=o
(*-O,
•j . v — ' • -• "j~Q>
F=0 ^ /
Differentiating (405), we get
€=00 (f-O2 (*~O2
de-* rdcj> —rf—+* _ / - ~~rf~+<t> dea\
T— = I -£- e de—[e — ± }
^ da^ J dc^ \ ddij
where ea denotes the least value of e consistent with the exter- nal coordinates. The last term in this equation represents the part of de~c jda^ which is due to the variation of the lower limit of the integral. It is evident that the expression in the brackets will vanish at the upper limit. At the lower limit, at which ep = 0, and eq has the least value consistent with the external coordinates, the average sign on ^]6 is superfluous, as there is but one value of A1 which is represented by — dea/dar Exceptions may indeed occur for particular values of the external coordinates, at which dejda^ receive a finite increment, and the formula becomes illusory. Such particular values we may for the moment leave out of account. The last term of (408) is therefore equal to the first term of the second member of (407). (We may observe that both vanish when n > 2 on account of the factor e$.) We have therefore from these equations
F=0
or
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 true for any value of «. If we diminish o>, the average value of the parenthesis at the limit when « vanishes becomes identical with the value for e = e'. 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 equa- tion as