268

DOC.

7

PROBABILITY CALCULUS

Published in Annalen

der

Physik 33

(1910):

1096-1104.

Received 29

August 1910, published

20

December 1910.

[1]Ludwig

Hopf

(1884-1939),

who had received

his

degree

with Sommerfeld in

1909, was

registered

in all

three of Einstein's

courses

at

the

University

of Zurich

in

the

summer

semester

1910.

He collaborated with Einstein

not

only on

this

paper

and the

subsequent

one (Doc.

8),

but also

helped

him

to identify

a

calculational

error

in

an

earlier

publication;

see

Einstein

1911e

(Doc.

14)

and the

notes to

this document. For evidence of their

personal relationship

at

the

time of their collaboration

on

the

present paper, see

also Einstein

to Ludwig

Hopf, 21

June

1910,

and Einstein

to Ludwig

Hopf,

2

August

1910.

[2]For the

use

of

a

Fourier

decomposition

of

the radiation

field in

the

study

of heat

radiation,

see,

e.g.,

the classic book

by

Planck,

Planck

1906,

in

particular pp. 118ff.

Einstein had earlier

reviewed this

book;

see

Einstein

1906f (Vol.

2,

Doc.

37).

He

quoted

it

in his

second

paper

with

Hopf;

see

Einstein and

Hopf

1910b (Doc.

8), p.

1107, fn. 2.

[3]This

definition of

probability is

different from

Planck's;

for

Planck's

definition,

see

Planck

1906,

p. 139;

for

a

historical discussion of the relevance of

this

difference, see

the editorial

note

in Vol.

2,

"Einstein's

Early

Work

on

the

Quantum

Hypothesis,"

pp. 137-139,

and Kuhn

1978,

in

particular pp.

182-187.

[4]Contemporary

applications

of the

equipartition

theorem

to

heat radiation

assumed–

either

explicitly

or

implicitly-the

statistical

independence

of the

Fourier

coefficients and

led to

the

experimentally

refuted

Rayleigh-Jeans

law;

for historical

accounts,

see

Klein,

M.

1970,

pp.

234-237,

and Kuhn

1978,

pp.

143-152.

This

assumption

also

corresponds to

Planck's

hypothesis

of "natural radiation"

("natürliche Strahlung");

for

its

description

by

Planck

in

terms

of the Fourier

coefficients, see, e.g.,

Planck

1906, p. 133,

and

for its

role

in Planck's

analysis

of

black-body radiation,

see

pp.

187ff.

For

Einstein's views

on

the statistical

properties

of radiation

as

distinct

from those

of Planck in

1910, see

Einstein's

unpublished response to

Planck 1910a

(Doc.

3).

[5]The

inference

expressed in

the last

sentence

later

gave rise

to

a

controversy

between

Einstein and Max

von

Laue;

see

Laue

1915a,

Einstein

1915b,

and Laue

1915b.

In

1924

Planck

considered the

question

as

still

unresolved;

see

Planck

1924.

This discussion

is

mentioned

in

Klein,

M.

1964, p. 16.

[6]The

mathematical

problem

treated

by

Einstein and

Hopf belongs

to

the tradition of

cen-

tral

limit

theorems, going

back

at

least

to

Laplace

(see, e.g.,

Stigler 1986, pp.

136ff)

and

actively

pursued

around the

turn

of

this

century,

in

particular

by

the Russian school

(see, e.g.,

Maistrov

1974,

pp.

208ff).

At

the time of the

present

paper, however, probability theory

was

not yet

a

mathematical

discipline

with

a

standard literature to which

physicists

would

commonly

refer

(see,

for

instance,

the

complaint

about

the

neglect

by

physicists

of results achieved

in

statistics

in Ehrenfest and Ehrenfest

1911,

pp. 86-87).

In the criticism of Einstein's and

Hopf's

result

mentioned in note

3,

however,

von

Laue did

refer to

the

recently published

German translation

of the

Russian textbook

on

probability theory

by

Markoff,

Markoff

1912

(see

Laue

1915a,

p. 855,

fn.

1).

[7]The

following argument

derives

a

differential

equation

for

a

diffusion

process

in

a

way

similar

to

the

one

Einstein had used earlier in his

analysis

of Brownian

motion; see

Einstein

1905k

(Vol.

2,

Doc.

16),

pp.

557-558.

[8]There

should be

a

minus

sign

in front

of

the

right-hand

side of

the

equation.

[9]The

right-hand

side

of

this

equation

should

be

continued

by

"...,"

since only

the first

two

factors

are

written

out.

[10]The

an

of the

right-hand

side

should

be

an.