DOC.

27

DISCUSSION OF DOC.

26

427

attributes

to

the

fact

that

in

the

overwhelming majority

of

cases a

state

Za

is

succeeded

by

a more

probable

state

Zb.

From

among

all

the

states

Zb,

Zb',

Zb",

etc.,

to which

Za

can pass

in

the

very

short time

t,

the

state

Zb

will

practically always

appear,

because

it

possesses an

enormously greater probability

than the

state

Za

and all

of

the

other

states

Zb',

Zb",

etc.

Thus,

the

apparently

unidirectional

succession

of

states

actually

consists

in states

of

ever greater probability

following

successively

upon

each

other.

But such

an

argument

gains

some measure

of

persuasive

power

only

when

one

has

made clear

what

is

to

be understood

by

the

"probability"

of

a

state.

If

a system

left to

itself

passes

in

an

endless

succession

through

the

states

Z1

...

Zl

(in

the

most

varied

sequences),

each

state will

possess a

definite

temporal

frequency.

There

will

be

a

fraction

z1

of

a very long

time

T,

during

which

the

system

will

be

in

x

the

state

Z1; if,

for

large

T,

-1

tends toward

a limiting value,

then

we

call this

the

probability

W1

of the

first

state,

etc.

Thus,

the

probability

W

of

a

state

is

conceived

as

the latter's

temporal frequency

in

a

system

left

to

itself

an

infinitely long

time.

From

this

point

of

view,

it

is noteworthy

that

in

the

overwhelming majority

of

cases,

if

one

starts

out

from

a

specific

initial

state,

there

will exist

a

neighboring

state which

the

system-if

left to

itself

an

infinitely long

time-assumes

more

often than

it will do

other

states.

But

if

we forgo

such

a

physical

definition of

W,

the

statement

that

in

the

overwhelming

majority

of

cases

the

system passes

from

one

state to

a

state

of

greater

probability

is

a

statement

devoid of

meaning,

or-if

one

has

set W

equal

to

an

arbitrarily

chosen

mathematical

expression-is

an arbitrary

assertion.

If

W

is

defined

in

the

manner indicated,

then

it follows

from the

very

definition that

a

system

left to

itself

in

an

arbitrary

state

(and

isolated

from

without)

must

assume,

in

the

majority

of

cases,

successive states

of

ever greater probabilities,

and from

this it

follows

that

W

and the

entropy S

are

connected

by

Boltzmann's

equation

S

=

k

log

W

+

const.

This follows

from the

circumstance

that the

probability

W-insofar

as

the character of

a

unidirectional

flow

of

events

is

maintained

at

all-must

always

grow

with

time,

and that

there

cannot

be

a

function

independent

of

S

that

has this

property

at

the

same

time

as

S.

That the connection between

S

and

W

is

exactly

the

one

given

in

Boltzmann's

equation

follows

from the relations

Stotal

=

E

S,

Wtotal =

R(W),

which

hold

for

the

entropy

and

probability

of

states

of

systems

composed

of

a

number

of

subsystems.

If

one

defines W in

the

manner indicated,

as

temporal

frequency,

then Boltzmann's

equation

contains

right away

a

physical

statement.

The

equation

contains

a

relation

between

quantities

that

are

observable

in

principle, i.e.,

the

equation

is

either

correct

or