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. i x

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 46, 47

Average value of total kinetic energy for any given configuration

or for the whole ensemble = % v 47

System of n degrees of freedom. Average value of kinetic energy,

for any given configuration or for the whole ensemble = f . 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).

$ = \odVldc 87,88

V q = extension-in-configuration below a limiting value of the poten-
tial energy (e ? ). fa = \o^dV q jd fq 89,90

V p = extension-in-velocity below a limiting value of the kinetic energy

(*). ^ p = lo S dV p jd p 90,91

Evaluation of V p 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 < + 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 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 q l , . . ,q n for the
(generalized) coordinates, qi , . . . q n for the (generalized) ve-
locities, and

for the moment of the forces. We shall call the quantities
F l9 ...F n the (generalized) forces, and the quantities p 1 . . . p n ,
defined by the equations

Pl = ^- t p 2 = ^, etc., (2)

dqi dq 2

where e p denotes the kinetic energy of the system, the (gen-
eralized) momenta. The kinetic energy is here regarded as
a function of the velocities and coordinates. We shall usually
regard it as a function of the momenta and coordinates,*
and on this account we denote it by e p . 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 de p /dq 1 , 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 dptterVative, in other words, if the expression (1)
j.s t an t exact differential, we may set

where e q 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 (e 3 ) may depend on other variables
beside the coordinates q 1 . . . q n . We shall often suppose it to
depend in part on coordinates of external bodies, which we
shall denote by a x , # 2 , etc. We shall then have for the com-
plete value of the differential of the potential energy *

de q = FI dq l . . F n dq n A 1 da^ A 2 da z 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=q l dp l . . . + q n dpn~Pidqi . . .

p n dq n A l da-i A 2 da z 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
q 19 . . . q n , either alone or with the coordinates a 1? a 2 , etc. of
certain external bodies. It is not necessary that they should
be derivable from a force-function. The external coordinates
a 15 a 2 , etc. may vary with the time, but at any given time
have fixed values. In this they differ from the internal
coordinates q 1 , . . . q n , 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

. . . dp n dq l . . . dq n , (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

dp l ... dp n dq lf . . dq n (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 (e p )
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 p l of a sys-
tem passing the limit p^, or the limit p/', or by the q l 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 p l passing the limit p^. It will be convenient, and it is
evidently allowable, to suppose dt so small that the quantities
^ dt, q l dt, etc., which represent the increments of p l , q l , 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 p l passes the limit p^ in the interval
dt are those for which at the commencement of this interval
the value of p 1 lies between p^ and p^ p dt, as is evident
if we consider separately the cases in which p l is positive and
negative. Those systems for which p 1 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 dt 2 ) may be found by substituting p l dt for p^' p^ in
(10) or for dp 1 in (11).
The expression

Dpi dt dp z . . . dp n dqi . . . dq n (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 . . . dp n dqi . . . dq n 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 dq l

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*

-dDd Pl ... dp n d Sl ... dq n , (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, p l , . . . p n , q l , . . . q n , 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 p l dt, q l 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 F lt ...F n are functions of q 1 , . . . q n and
a lf a 2 , etc., which last are functions of the time, is analytically equivalent
to the condition that F lf . . . F n are functions of qi, ...q n and the time.
Explicit mention of the external coordinates, a 1? 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, A lt A 2 , 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.

.dp n d qi ... dq n (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 dp l . . . dp n dq^ . . . dq n

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 . . . dp n dq l . . . dq n ,

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% = D s ". 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 g 1 ^ . . ,q n ^ Q lt . . . Q n are two
systems of coordinates, and Pi, p n > P\i - P n the cor-
responding momenta, we have to prove that

J'...Jdp 1 ...dp n d qi ...d qn =j*...fdP l ...dP n dQ 1 ...dQ n ,(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

>P n >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(Q l9

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 dQ r /dp y is function
of the <?'s alone, we get *

* The form of the equation

d de p _ d df p
dp y dQ x dQx dp v

in (27) reminds us of the fundamental identity in the differential calculus
relating to the order of differentiation with respect to independent variables.
But it will be observed that here the variables Qx and p y 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 dQ x dp y

^^ n /^dQ L \ = _d_de, == d^
dQ x r^i W& %J d& cZft, d& '

But since f'

r i \ a (j/ r /

d -k = ^. (28)

*& ^0.

Therefore,

...g n )

... Q n )
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 . . . dp n dqi . . . dq n = I ... I dpj . . . dp n f dqi r . . , dq n ' } (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 ,...q n ) _ d( Pl ,...q n )

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 , ...q n ) _ d( Pl " 9 . . . q n ") d_ d( Pl , ...q n )

Now since the time if' is entirely arbitrary, nothing prevents
us from making if 1 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>, . . . q n ) dp," ' dp n " dqj* ' dq n

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(p lt ...

dtd( Pl ,...y n ")

which substituted in (34) will give
d

_
-

... n _

dtd( Pl ',...q n ')

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, p v q v 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, ...g n ) d(pi', . . . q n r )

d(a, ...h) ' d(p l f , . . . q n ') 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

" * ... dq n (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

Pd Pl ...dq n (40)

expresses the probability that an unspecified system of the
ensemble will be found in the element of extension-in-phase
dpi . . . dq n . 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 Ne 1 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, q n ' at the time ^, and P" for that of the phase
jp/', . . . q n " at the time tf',

2

18 CONSERVATION OF

J. . . JV dtf . . . dqj =f. . . Jp" dp{' . . . dq n ", (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. . .fd Pl > <%" =

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

P 1 ' = P f .

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...fpd Pl ...dq n = 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 c t and the unit of energy is multiplied by c e , all
indices of probability relating to systems of n degrees of

freedom will be increased by the addition of

"--
n log c t + 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* d Pl ' . . . 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' Q te"> 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 F 1 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

d Pl > . . . 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 P 1 / , 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 F n is a quadratic function of the differences
Pi ~ p i"> <?i" - Ci", etc., the phase (P x ", 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 F 1 , 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 (7 2 , 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 t f shall give the function F' the same value as the corre-
_ sponding coordinates and momenta at the time t n give to F".
Therefore F n 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 F 1 , 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" = k 9 (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 t f 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 U r We have
then by definition

F=k

l ...dq n , (58)

24 CONSERVATION OF EXTENSION-IN-PHASE
Fk

F=l

But since F is a homogeneous quadratic function of the
differences

we have identically

F=k

rt

d(pi -Pi) . . . d(q n - Q n )
kF=k

rwy&i

F=l

-Pj...d(<!.-Q 1 ).

That is U=k n U l} (61)

whence dU= U 1 nk n ~ l dk. (62)

But if k varies, equations (58) and (59) give

F=k-\-dk

dU = I . . . I dpi . . . dq n (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~ k d U = C Ui n e~ k k n ~ 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--=k B= 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
r x . . . r 2n 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 r l , . . . r 2n 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 r l , . . . r 2n , 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^ , . . . r 2n , leaving b as
well as a to be determined. Our 2 w 2 finite equations en-
able us to regard all the variables r^ , . . . r 2n , and all functions
of these variables as functions of two of them, (say r l and r 2 ,)
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 , r 2 ) c?7* 2 7 dTi . .

whence -^7 TT- db ^- dr^-\- -= r 2 . (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,r z ) d(r^ ...r Zn ) d(c, ... h)
d(a,b) " d(a,...h) d(r 99 ...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 r l and r z in terms of 'r^ , . . . r 2n .
The integral equations already obtained give <?,... h and
therefore the Jacobian d(c, . . . A)/c?(r 3 , . . . r 2n ), in terms of
the same variables. But in virtue of these same integral
equations, we may regard functions of r 19 . . . r 2n as functions
of r l and r% with the constants c, . . . h. If therefore we write
the equation in the form

d(ri, . . .r 2n ) _ r 2 r i ,

' ~ **- ..h) dr *> (77)

d(r s , ..r 2n ) d(r 8 , . . . r 2n )

the coefficients of dr l and dr% may be regarded as known func-
tions of r x and r 2 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(r 8 , . ..r an ) d(r 8 , ...r 2n )

which may be integrated by quadratures and gives V as func-
tions of r 1? r 2 , ...<?,... A, and thus as function of r 1? . . . r 2n .
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

r 2 dr r^ dr z = 0,

and r and r 2 are given in terms of r v . . . r 2n by the differential
equations of motion. When we have obtained 2 n 2 integral
equations, we may regard r 2 and r^ as known functions of r l
and r 2 . 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.,...r fc )

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 db f 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 r v . . . r 2n , 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 r v . . . r 2w , and
t, leaving a single constant (a) to be thus determined. Our
2 % 1 finite equations enable us to regard all the variables
r v . . . r 2n as functions of a single one, say r r

For constant values of 5, ... A, we have

**-* + ft* (82)

Now

* * \MI 1 , _

-5 da dr* . . . dr zn =

t

da . . dh

d(a, ...h)

^"^ I f "

J J d(a } ... A) d(r 2 , . . . r 2n )

where the limits of the integrals are formed by the same
phases. We have therefore

^' A >, (83)

da " d(a,...h) d(r t , . . . r, n )
by which equation (82) may be reduced to the form

da =

M M
a, . . . h) d(b, ... A)

d(r 2 , . . .

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, ...r 2n ) d(r 2 , ...r 2n )

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(r z , . . . r zn )

for the integration of the equation

dr, -r l dt = 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
dq 1 dp l dp l dq l )~

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,... dq n . (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 dp l . . . dq n (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 ,

* 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 q v . . .q m , p v . . . p m be the
coordinates and momenta of A, and q m+l , . . . q m+n , p m+l , . . . p m+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^ . . . <?,+, p v . . . p m+n .
The probability that the phase of the system (7, as thus defined,
will fall within the limits

dpi , . . . dp m+n , dq 1 , . . . dq m +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 e A + 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 e AB . The energy of the system C is now
A + B + ABI an d 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 e AB (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 rj r and 77" be the indices of probability in two independ-
ent ensembles which are each in statistical equilibrium, then
rf _j_ y w ni 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 + if 1 will be a con-
stant of motion. Now when infinitesimal forces are added to
the compound systems, if r/ + rf 1 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, O x , O 2 , 11 3
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^ + a3L-, (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 + w 3 '
will remain constants of motion, and a function differing in-
finitely little from e l + 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 (e q ) depends not only upon the coordinates q l . . . q n
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 de q jda v de q fda 2 , etc., while de q jdq v
... de q /dq n 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 , . . . q n , p\ , . . . p n , 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, a x ,
2 , 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 /

, \

(- I ^ + I ) = p

all

phases

all Jf

-/^ e ~ d Pi d v- ~ ete -> ( 106 )

phases
t

or, multiplying by e, and setting

-^ = ^ - = ^ etc ->

all

|d = ^ f. . .f

ee

phases

i e dp l . . . dq n

phases

r r

i I . . .

phases

r * ( fcf

2 J ...JA 2 e & dp l ...dq n + 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 & d Pl ... dq a . (108)

phases

Comparing this with the preceding equation, we have

<ty = d - ~ d - A! da^ - 2 2 da 2 - etc. (109)

(jj) (y

Or, since fe J = ,, (110)

and ^=^

d\f/ = yd AI da,i >Z 2 d2 etc.
Moreover, since (111) gives

dty - c?e = cfy + ^, (113)

we have also

dk drj ^ ddi A 2 da 2 etc. (114)

This equation, if we neglect the sign of averages, is identi-
cal in form with the thermodynamic equation

de + A l da 1 + A z da z + etc.
drj= y -, (115)

or

de = Td-rj A! da L A z da 2 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 ] dTA l da l A z da< 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 . . . dx v dy v dz v ,

and for the product of the differentials of the momenta
m l dxi mi dyi m^ dz 1 . . . m v dx v m v dy v m v dz v .

The product of these expressions, which represents an element
of extension-in-phase, may be briefly written

mi dxi . . . m v dz v dxi . . . dz v ;
and the integral

e @ mi dxi . . . m v dz v dxi . . . dz v (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

=e & e 2> 20 . (120)

The potential energy (e 3 ) 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

m v dz v . (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 4 V>, <& = 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 x 19 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

e p = ! = 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& d Xl . (125)

The expression becomes more simple when the velocity is
expressed with reference to the energy involved. If we set

s=(^x l ,

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, s 2 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 . . . /**.** . . . dz v (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 e p 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 e p 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 d Pl ' ' d Pn

_oo J oo api

-=r- (129)

e dpi . . . dp n
Now we have by integration by parts

ty-C

r PI <^~^- d Pl = 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 dp 1 . . . dq n for dp . . . dp n 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 e q 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 ^ + F l = - + F l

dqi dqi dqi

The work done by the force F may be evaluated as follows :

r r d *

= I Pi dq t -f I ydqit
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 dp t =. 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 p 1 in the phase originally considered,
and the quantities before the integral sign may be taken as
relating to that phase. We have therefore

i = i pl ^L t (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 F l , . . . F n
under the conditions mentioned.

The following transformation will not only give the value
of the average kinetic energy, but will also serve to separate
the distribution of the ensemble in configuration from its dis-
tribution in velocity.

Since 2 e p 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

2e p = < 2 + w 2 2 ... + iv 2 (133)

where MJ . . . u n 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 dp 1 . . . dp n

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 2e p 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 dp x jdu y for the general element
of the Jacobian determinant. We have

ENSEMBLE OF SYSTEMS. 53

dp x d de d r=n de du r

du y du y dq x du y r \ du r dq x

r ?" ( ^ e du r \ d de _ du y

Therefore

d(p, ...p n ) __d(u, .. . Q
d(u, . . . u^) d(q, . . . q n )

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 , . . . q n . 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 A p .
If we set

e & = I . . . / e A p dp,...dp n

+00 +00 Mj 2 . . . n 2

f. . . C

e 20 d Ul . . . du n = (27r) , (140)

and *, = * - fe (141)

* It will be observed that the proof of (137) depends on the linear relation

du r

between the u's and q's, which makes constant with respect to the differ-

dq x

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

dq l . . . dq n , (142)

where the constant ty q 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 e a 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 . . . q n ' the values of qi , . . . q n which give f q its least value
e a , it is evident that e g e a is a homogenous quadratic function of the differ-
ences ?! qi, etc., and that dq t , . . . dq n 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! . . . dp n ,

00 CO

for which we have found the value A p * (2 TT 9) 3. By the same method, or
by analogy, we get

where A 9 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 A p 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~ e a

J ' ' -f ( Q f a)e dq l . . . dq n

00 00

+00 +eo * a

J . . .J e dqi . . . dq n

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

d Pl ...dp n ,
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^ . . . du n , (143)

The evaluation of this expression is similar to that of

+00 +00 _!?

...s f e & dp l ...dp n

+00 +00 _ C JL
f...fe & d Pl ...dp n

- 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 = 2 na
Adding the equation

*i> = 2 ne >
we have I e a = n e.

\

\

56 AVERAGES IN A CANONICAL ENSEMBLE.

/ ^p-fp
je & **d Pl ...dp n , (144)

or again

r r^=^ i

I . . . / e < Ar^Ti 4i ( 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

. dp n dqi . . . dq n , (146)

may also be expressed in the form

. . dq n dq l . . . dq n . (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 ty g .
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

^ ...dg n (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(P l9 . . . P.) <%i . . . q n ) d(q l9 ...q n )

..-) d(Q l9 ...Q n )

where q l , . . . q^ft , . . .p n and Q l , . . . Q n9 P 1 , . . . P n are two
systems of coordinates and momenta.* It follows that

i>

= r

J

* See equation (29).

AND EXTENSION IN VELOCITY. 59

and

/Cf d (Ql, ... Qn)\% JT> Jp

' ' J \d(P^ ~^P}) ' *

"'<%>!... <jp.

= c.

J

>!,-.. W

The multiple integral

>! . . . dp n dqi . . . rf^, (151)

which may also be written

1 . . . dq n dqi . . . dq n , (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 . . . dq n , (153)

which is equivalent to

A-^! . . . dq n dq t . . . dq n , (154)

an element of extension-in-phase. We may call

A^ ...dq n (155)

an element of extension-in-configuration, and

. . . dp n , (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 p l q 1 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 q 1 . . . q n , and the
number of systems of which the coordinates fall between the
limits q 1 and q l + dq l , . . . q n and q n + dq n is expressed by

D.A^Zi *2n, (158)

AND EXTENSION IN VELOCITY. 61

D q 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^dq t . . . dq n . (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.

. . . dg n . (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 D q 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 q lt ...q n 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,...p n the same in each. This again would signify nothing
independently of the system of coordinates employed.

62 EXTENSION IN CONFIGURATION

within certain infinitesimal limits of velocity. The second
of these numbers divided by the first expresses the probability
that a system which is only specified as falling within the in-
finitesimal limits of configuration shall also fall within the
infinitesimal limits of velocity. If the limits with respect to
velocity are expressed by the condition that the momenta
shall fall between the limits p 1 and p 1 + dp l , . . . p n and
Pn + dpm the extension-in-velocity within those limits will be

. . . dp n ,
and we may express the probability in question by

e^\^d Pl . . . dp n . (162)

This may be regarded as defining rj p .

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 . . . dp n , (163)

or its equivalent

J 1 . . .J**4Mb . . . dg n , (164)

taken within the given limits.

It follows that the probability that the system will fall
within the limits of velocity, ^ and ^ + dq 19 . . . q n 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
e 1 ?; therefore the value of TJ P must be independent of the
system of coordinates and momenta. We may call e 1 ? the
coefficient of probability of velocity, and tj p the index of proba-
bility of velocity.

AND EXTENSION IN VELOCITY. 63

Comparing (160) and (162) with (40), we get

eV* = P = e l (166)

or rj q + I P = ^. (167)

That is : the product of the coefficients of probability of con-
figuration and of velocity is equal to the coefficient of proba-
bility of phase; the sum of the indices of probability of
configuration and of velocity is equal to the index of
probability of phase.

It is evident that e 1 * and e 1 ? 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 c t , and the unit of energy by
c e , every rj q will be increased by the addition of

n log c t + i?i log c. , (168)

and every rj p 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 $t n .

In demonstrating this theorem, we shall write q^ . . . q n f 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(j ll ...^) = 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, V l , . . . V n of which a
system in a given configuration is capable. We may conceive
of the system as having a certain velocity F~ combined with a
part of each of these velocities V l . . . V n . By a part of V\ is
meant a velocity of the same nature as V\ but in amount being
anything between zero and V r 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\ , . . . V w 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 , . . . V n
taken separately.

The more general case may be reduced to this simpler case
as follows. The velocity F 2 may always be regarded as
composed of two velocities Vj and V 2 ", of which VJ is of
the same nature as V l , (it may be more or less in amount, or
opposite in sign,) while V 2 " satisfies the relation that the

5

66 EXTENSION IN CONFIGURATION

kinetic energy due to V l and V 2 n combined is the sum of the
kinetic energies due to these velocities taken separately. And
the velocity V B may be regarded as compounded of three,

*Y F 3"> *Y" of which v * is of the same nature as F i ' V *
of the same nature as V 2 ", while V B " 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 , . . . V n 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 e 12 for the kinetic energy of the velocities F x 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 F 2 . Let the
mutual energy of every pair of the velocities Fj , . . . V n be
expressed in the same way. Analogy would make e n represent
the energy of twice V 1 diminished by twice the energy of Fi ,
i. e.y e n would represent twice the energy of Fi , although the
term mutual energy is hardly appropriate to this case. At all
events, let e n have this signification, and e 22 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 dq t , 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 " ^d qi ...dq n} (179)

u

conflg.

where the integrations cover all possible configurations. The
value of i|r g is evidently determined by the equation

r ^ r _!?

=J . . .J e %*dfc . . . dq n . (180)

e

config.

By differentiating the last equation we may obtain results
analogous to those obtained in Chapter IV from the equation

- *" ~ f

J * J & &dPl ' ' '

e

.

phases

As the process is identical, it is sufficient to give the results :
dfa = rj q d J^i J^da^ etc., (181)

AVERAGES IN A CANONICAL ENSEMBLE. 69
or, since \f/ q = 7 g + ^ g , (182)

and <fy c = < g 4- ^ g rf + <fy a , (183)

ckg = cfyg ^etai J 2 ^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 ^r q is known
as function of S and x , a 2 , etc., we can obtain by differentia-
tion e or e q , and A ly A Zy etc. as functions of the same varia-
bles. We have in fact

* = * f