32

THEORY OF

THERMAL

EQUILIBRIUM

[7]

system an

additional condition of the kind

u(P1,...,qn)

=

const.

that

cannot be

reduced

to

the

form

p(E) =

const.,

then it

would obviously be

possible to choose

initial conditions

such

that

each of

the

N systems

could

have

an

arbitrarily

prescribed

value for

p.

However,

since

these values

do

not

vary

with

time, it

follows,

e.g.,

that

for

a

given

value

of

E

any

arbitrary value

might

be

assigned to

S(p,

extended

over

all

systems,

through

appropriate

selection of initial conditions.

On

the other

hand,

Yjp

is

uniquely

calculable

by

the distribution of

states,

so

that other distributions

of

states correspond to

other

values

of

Yip.

It is thus clear that the exis-

tence

of

a

second such integral

(p

would

necessarily

have

the

consequence

that

the

state

distribution

would

not be

determined

by

E

alone but

would

necessarily have

to

depend

on

the initial

state of

the

systems.

If

g

denotes

an

infinitesimally

small

region

of

all

state

variables

P1,...Pn,

q1,...qn,

which

is

chosen such

that

E(p1...qn)

lies

between

E

and

E

+

SE

when

the

state

variables

belong

to

the

region

g,

then the

distribution

of

states

is characterized

by an

equation

of the

form

[8]

dN

=

iKpj,...,?

)

dpv..dq

,

J9

where

dN

denotes the

number

of

systems whose state

variables

belong

to

the

region

g

at

a

given

time.

The

equation

expresses

the condition that the

distribution is

stationary.

We now

choose such

an

infinitesimal

region

G.

The

number

of

systems

whose

state

variables

belong

to

the

region

G

at

a

given

time

t

=

0

is then

dN

=

*{Pv...q)

dP*

•• •

dQ

G

1 n

where the

capital

letters

indicate that the

dependent

variables

pertain

to

time

t

=

0.

We now

let

elapse

some

arbitrary time

t.

If the

system

possessed

the

specific state

variables

P1,...Qn at

time

t

=

0,

then it will

possess

the

specific state

variables

P1,...,qn

at

time

t

=

t.

Systems

whose state