50

FOUNDATIONS OF THERMODYNAMICS

physical

system,

i.e.,

of

a

system

that

assumes

a

stationary

state,

then for

each

region

T

the

quantity r/T has

a

definite

limiting

value for

T

= od.

For

any

infinitesimally

small

region

this

limiting

value is

infinitesimally

small.

The following

consideration

can

be based

on

this postulate.

Let

there

be

very

many

(N)

independent

physical

systems,

all

of which

are

represented

by

the

same

system

of equations

(1).

We

select

an

arbitrary

instant

t and

inquire

after the distribution

of

the

possible states

among

these

N

systems,

assuming

that the

energy E

of

all

systems

lies

between

E*

and

the

infinitesimally

close value

E*

+

6E*. From

the postulate introduced

above,

it follows

immediately

that the

probability

that the

state

variables

of

a

system randomly

selected

from

among

N systems

will lie within the

region

T

at

time

t

has

the value

lim

T

=

const.

T

=

00

The number

of

systems whose state

variables lie within

the

region

T

at

time

t

is thus

N

•

lim

T/T,

T

=

oo

1

i.e.,

a

quantity

independent

of

time.

If

g

denotes

a

region

of the

coordi-

nates

p1...pn

that is

infinitesimally

small in all variables,

then the

number

of

systems

whose

state

variables fill

up an

arbitrarily

chosen

infinitesimally small

region

g

at

an

arbitrary

time will

be

[5]

(2)

dN

=

e(p1...pn)

fg

dp1...dpn

.

The

function

e

is obtained

by

expressing

in

symbols

the

condition that

the distribution

of

states expressed

by

equation

(2)

is

a

stationary

one.

Specifically, the

region

g

shall

be chosen such

that

p1

shall lie

between

the definite values

p1

and

p1

+

dp1,

p2

between

p2

and

p2

+

dp2,...Pn

between

pn

and

pn

+ dpn;

then

we

have at

the time

t

dNt

=

e(p1...pn).dpl.dp2...dpn,