74

DOC.

3

THEORY OF THERMAL

EQUILIBRIUM

1898a,

§

29, p. 82).

It

is

implicit

in Boltzmann

1871a, and

it

appears explicitly

in Boltzmann

1885,

pp.

78-79,

where Boltzmann first defined

the term

"Ergode" (see

note

6),

and

in Boltz-

mann

1898a,

§

32,

p.

89. The

striking

locution,

"unendlich

viele

(N) Systeme" ("infinitely

many

(N)

systems")

may

have been taken from

Boltzmann; exactly

the

same

words

are

used,

e.g.,

in Boltzmann

1887, p.

207.

[5]

Vi

denotes the internal

potential energy

of

the

system.

[6]

This condition

on

the

energy

is

implicit

in

Boltzmann 1871a,

p.

707, and its

significance

is

discussed

in Maxwell

1879, pp.

548-560.

It

is

employed

explicitly

in Boltzmann

1885,

pp.

78-

79, to define the

concept

of

an

"Ergode,"

a

vir-

tual

ensemble of

systems

with identical ener-

gies,

for which the

energy

is

the

only explicitly

time-independent

conserved

quantity.

The defi-

nition

of

an "Ergode"

in Boltzmann 1898a,

§

32,

p.

89,

does not include this condition.

[7]

The same

notation for constants

of

motion

is employed on pp.

78-79 of

Boltzmann

1885,

where the

concept

of

an

"Ergode" (see note 6)

was

first defined.

In Boltzmann 1898a the

same

notation is used,

more

generally,

to

represent

the

integrals

of

motion

of

a

system (see

§

30),

but it

is

not used in the discussion

of

an

"Ergode"

(§

32).

[8]

In Boltzmann 1898a,

§

25, uppercase

Pv

and

Qv

are

used

to

represent

coordinates and

mo-

menta

at

t

=

0,

with lowercase

pv

and

qv

representing

coordinates and momenta at

t

=

t;

G and

g are

used there to

designate

the associ-

ated

regions

of

phase space.

[9]

This

equation

should be

dN

= (px,

...

qn)

T

dpx

. . .

dqn.

J

8

[10]

Boltzmann

1898a.

[11]

The

idea

of

a

virtual ensemble

of

compos-

ite

systems

is

anticipated

in Boltzmann

1871a,

pp.

707-711, and in Boltzmann

1898a,

§

35,

where Boltzmann

considered

an

ensemble

of

systems,

each

of

which

is

divided into two

parts

separated

by

a

heat-conducting

wall.

[12]

The

symbol

H

represents

the

uppercase

Greek eta.

[13]

Cf. Boltzmann 1898a,

§

37,

p.

108, eq.

(115).

The constant h

was

first introduced

by

Boltzmann

in Boltzmann

1868,

p.

523,

and used

regularly

thereafter; see, e.g.,

Boltzmann

1896,

§

7, p.

48.

[14]

8E

should be 8E.

[15]

Such

a

distribution

is

now

called

a canon-

ical distribution,

following

Gibbs

1902,

pp.

33-

34. The

concept

of

such

an

ensemble

is

already

implicit

in Boltzmann

1872, pp.

368-370.

For

a

discussion

of Einstein's

later

adoption

of

Gibbs's

terminology,

see

the editorial

note,

"Einstein

on

the Foundations

of

Statistical

Physics,"

pp.

54-55.

[16]

Boltzmann referred

to

Kirchhoff

as

the

source

of

such

a probabilistic interpretation

of

this distribution

(see

Boltzmann 1898a,

§

38,

p.

112,

and

§

37).

Boltzmann

was

probably

refer-

ring

to

Kirchhoff 1894,

Lecture

13,

pp.

134-

141.

[17]

The limits

of

integration are

defined

by

((x')

taking a

value between

y'

and

y'

+

A,

where

y'

=

y/a2.

[18]

In

fact,

a

must

be

greater

than

1.

[19]

The

inequality

should

be

2

d[u(E1)] w(E2).

[20]

Einstein's

proof

of

what Paul Hertz

dubbed the

"Trennungssatz"

(the

assertion that

after

a composite system

Z

is

separated

into two

parts,

Z1

and

Z2,

h

=

h1

= h2)

is

strongly

criti-

cized in

Hertz,

P.

1910a, pp.

247-255.

Hertz

argued that,

in contrast to the

proof

of the

"Ver-

einigungssatz"

(the

assertion

that,

after two

sep-

arate

systems,

Z1

and

Z2, are

joined to

form

a

composite system

Z,

h1 = h2 =

h),

the

proof

of

the

"Trennungssatz"

requires

the

assumption

"that

all

phases

of

the

composite system are ex-

plored"

("daß alle Phasen des

zusammenge-

setzten

Systems

durchwandert werden")

(Hertz,

P. 1910a,

p.

254).

In his

reply

to

Hertz,

Einstein

wrote:

"The

...

objections

against

an

obser-

vation

on

thermal

equilibrium

contained in

my

first

essay

on

the

subject

rest

upon a

misunder-

standing

that

was brought

about

by an

all

too

terse

and

insufficiently

careful formulation of

that

observation"

("Die

...

Bemerkungen ge-

gen

eine in meiner

ersten

einschlägigen

Abhand-

lung

enthaltene

Betrachtung

über

das Tem-

peraturgleichgewicht

beruht

auf

einem

Mißverständnis,

das durch eine allzu

knappe

und nicht

genügend sorgfältige Formulierung je-

ner

Betrachtung hervorgerufen

wurde")

(Ein-

stein

1911c,

p.

175;

see

also

Einstein to Paul

Hertz,

14

August

1910).

Einstein did not

explain

the nature

of

the

misunderstanding.

For another

discussion

of

this

topic by

Einstein,

see

Einstein

1903

(Doc.

4),

§

4. See also Hertz, P.

1910b,

as

well

as

Ornstein

1910, 1911,

for

further discus–