FOUNDATIONS

OF STATISTICAL

PHYSICS

53

distribution and the

equipartition theorem,

Einstein

gave a more general

derivation of

his

expression

for

the

entropy,

which

no longer requires

the distinction between kinetic

and

potential energy

and

is

supposed

to be valid for

nonequilibrium

distributions.

The

paper

concludes with

a

derivation

of

the second law that also claims

validity

for

the

nonequilibrium case.

However,

the derivation

depends upon

the

assumption

that

"ever

more probable

state distributions will follow

more improbable

ones"

("immer

wahrscheinlichere

Zustandsverteilungen

auf

unwahrscheinlichere

folgen werden");[64]

here,

the term

"state distributions"

("Zustandsverteilungen")

refers to the distribution

of

microstates

of

a

virtual ensemble

of

systems over a hypersurface

of

constant

energy.

Ein-

stein

was

later criticized for his failure

to

prove

this

assumption,[65]

but his next

paper

modifies it in

a way

that makes

clear

that he did not intend to

deny or

to

ignore

thermo-

dynamical

fluctuations.[66] The main

difficulty

with which Einstein

was

grappling concerns

the kind

of

microstate

or

state distribution that should be related

to the

thermodynamic

state

of

a

system

in

attempting a

derivation

of

such

thermodynamical

principles as

the

second

law.[67]

On

6

April

1904 Einstein

wrote to his friend

Marcel Grossmann that he

was sending

him

a

paper

he had

just

submitted to the

Annalen, in

which he treated

"the

atomistic

theory

of

heat without the kinetic

hypothesis"

("die atomistische Wärmelehre ohne die kine-

tische

Hypothese").

Einstein's

last

paper

devoted

exclusively

to the foundations

of

statis-

tical

physics,

Einstein 1904

(Doc. 5),

was

submitted for

publication

in

late March and

appeared

in

early

June 1904. This

paper

is

the culmination

of Einstein's

efforts to

gener-

alize and extend the foundations

of

statistical

physics.

It

is

distinguished

from his two

previous

papers by a more polished

and direct

style,

and

by

the concrete and novel

physical

consequences

drawn from foundational

investigations,

which foreshadow

Einstein's

later

employment

of

the

principles

of

statistical

physics

in such fields

as black-body

radiation,

specific

heats,

Brownian

motion,

and critical

opalescence.

Most

of

the

paper's

formal

apparatus

is

taken from the

previous paper,

but there

are a

number

of

new

results.

Among

these

are

(1)

an improved

derivation

of

the second

law,

replacing

the

problematic assump-

tion about the

monotonic increase

in

the

probability

of

a

distribution of

microstates with

the

assumption

of

a

monotonic increase

in

the

probability

that the

energies,

E1,

E2,

. . . ,

El,

of

the individual members

of

a

set

of

systems

will each lie

in

a given

range,

E1,

to

E1,

+

8E1,

of

energies;

such

a

distribution

of

energies

is

now designated as

the

"state" of

the

set

of

systems;

(2)

a

determination

of

the constant

K

(K

=

k/2,

where

k is

Boltzmann's

constant),

which establishes

a

relation between the values

of

Avogadro's

number

and

of

Boltzmann's

constant. Most

important,

however,

are

results that

anticipate some

of

[64]

Einstein 1903 (Doc. 4), p. 184.

[65]

See Hertz, P. 1910a, p. 552. Einstein

re-

plied to Hertz in Einstein 1911c. See also

Ein-

stein 1902b (Doc. 3), note 20, and Einstein 1903

(Doc. 4), note

17.

[66]

See Einstein 1904 (Doc. 5), pp.

355-357.

For a more complete discussion

of

the questions

posed by this assumption, see Einstein 1903

(Doc. 4), note

17.

For

Einstein's

later views on

the statistical character of the second law, see,

e.g., Discussion/Einstein

1911, pp.

436-443,

and

Einstein

1915a, pp.

262-263.

[67]

See Hertz, P. 1916, pp.

547-558,

for

a

discussion

of

the relations between microstates,

phenomenological states, and entropy informed

by

Einstein's

views.