232

DOC.

9

CRITICAL OPALESCENCE

[6]

W

is

commonly

equated

with

the number of different

possible ways (complexions)

in

which

the

state

considered-which

is incompletely

defined

in

the

sense

of

a

molecular

theory

by

observable

parameters

of

a

system-can

conceivably

be realized.

In order

to

be

able to calculate

W,

one

needs

a complete

theory (perhaps

a

complete

molecular-mechanical

theory)

of the

system

under consideration.

Given this kind

of

approach,

it

therefore

seems

questionable

whether Boltzmann's

principle

by

itself

has

any

meaning whatsoever,

i.e.,

without

a complete

molecular-mechanical

or

other

theory

that

completely represents

the

elementary

processes

(elementary

theory).

If

not

supplemented

by an

elementary theory

or-to

put

it

differently-considered

from

a

phenomenological

point

of

view,

equation

(1)

appears

devoid

of

content.

However,

Boltzmann's

principle

does

acquire some

content

independent

of

any

elementary theory if

one assumes

and

generalizes

from

molecular

kinetics

the

proposition

that the

irreversibility

of

physical

processes

is

only

apparent.

For let the

state

of

a system

be determined

in

the

phenomenological

sense by

the

variables

A1

...

Xn

that

are

observable

in

principle.

To each

state

Z there

corresponds

a

combination of

values

of these

variables.

If the

system

is

externally

closed,

then the

energy-and,

indeed,

in

general,

no

other function of the

variables-is

constant.

Let

us

think of

all

the

states

of the

system

that

are

compatible

with

the

energy

value

of the

system,

and let

us

denote them

by

Z1...Zl.

If

the

irreversibility

of the

process is

not

one

of

principle,

then,

in

the

course

of

time,

the

system

will

pass

through

these

states

Z1...

Zl

again

and

again.

On

this

assumption, one can speak

of the

probability

of the

individual states in

the

following sense:

Suppose

we

observe

the

system

for

an immensely

long

period

of time

0 and

determine the fraction

X1

of the

time 0

during

which

the

system

is

in

the

state

Z1;

then

x1/0

represents

the

probability

of the

state

Z1.

The

same

holds

for the

probability

of the other

states

Z.

According

to

Boltzmann,

the

apparent

irreversibility

must

be attributed

to

the

fact

that the

states differ in

their

probabilities,

and that

the

system

is

probably

going

to

assume

states

of

higher probability,

if it

happens

to find

itself

in

a

state

of

relatively

low

probability.

That

which

appears

to

be

completely

law

governed

in

irreversible

processes is

to be

attributed

to

the

fact

that

the

probabilities

of the

individual states

Z

are

of

different

orders

of

magnitude,

so

that

a given

state

Z

will

practically always

be

followed

by

one

state,

from

among

all

the

states

bordering on

Z, because

of

this

one

state's

enormous probability as compared

with

the

probabilities

of the other

states.

It

is

this

probability

we

have

just

described,

for the definition of

which

no

elementary

theory

is needed,

which

is

related

to

the

entropy

in

the

way

expressed

by

equation

(1).

It

can

easily

be

recognized

that

equation

(1)

must

really

be valid

for the

probability

so

defined. For

entropy

is

a

function that does

not

decrease

in

any

process

in which

the

system

is

isolated

(within

the

range

of

validity

of

thermodynamics).

There

are

other

functions, too,

that

have this

property; however,

if

the

energy

E

is

the

only

function of

the

system

that

does not

vary

with

time,

then

all

of these

functions

are

of the

form

p(S,

E),

where

dy/dS

is

always positive.

Since

the

probability

W

is,

as

well,

a

function

that does

not

decrease

in

any

process,

then

W

is

also

a

function of

S and

E

alone,

or-if

only

states

of the

same energy are being

compared-a

function of

S

alone.

That

the relation between

S

and

W

given

in

equation

(1)

is

the

only possible one can

be