48

FOUNDATIONS OF THERMODYNAMICS

Doc.

4

A

THEORY OF

THE

FOUNDATIONS

OF THERMODYNAMICS

by A.

Einstein

[Annalen

der

Physik 11

(1903):

170-187]

[1]

In

a

recently

published

paper

I

showed

that the

laws of

thermal

equi-

librium

and

the

concept

of

entropy

can

be derived

with

the

help

of the kinetic

theory

of heat.

The

question

that then arises naturally is

whether

the

kinetic

theory

is

really

necessary

for the derivation

of

the

above

foundations

of

the

theory

of

heat,

or

whether perhaps assumptions

of

a more

general

nature

may

suffice. In this article it shall

be

demonstrated that the latter is the

[2] case,

and

it shall

be

shown

by

what

kind

of

reasoning

one can

reach

the

goal.

§1.

On a

general

mathematical

representation

of

the

processes

in

isolated

physical

systems

Let

the

state

of

some

physical

system

that

we

consider

be uniquely

determined

by

very

many

(n)

scalar quantities

P1,P2...pn,

which

we

call

[3]

state

variables.

The

change

of

the

system

in

a

time

element dt

is then

determined

by

the

changes

dp1,dp2...dpn

that the

state

variables

undergo

during

that time

element.

Let

the

system

be

isolated, i.e.,

the

system

considered should

not

interact with other

systems.

It is then clear that the

state of

the

system

at

a

given

instant of time

uniquely

determines the

change

of

the

system

in

the

next

time

element

dt, i.e., the

quantities

dp1,dp2...dpn.

This

statement

is

equivalent to

a

system

of

equations

of the

form

dpi

(1)

-jj-

=

V^PyPj

(i

=

1

...

i

=

n)

,

where

the

Q's

are

unique

functions of their

arguments.

In general,

for

such

a

system

of linear differential

equations

there

does not

exist

an

integral of

the

form