214

BROWNIAN MOTION

Vt2

=

2R

N

(II)

where A

is

the

change

in the

parameter

a,

and

B

is

what

Einstein calls the

mobility

of

the

system

with

respect

to

a. By considering

the effect

on

the motion

of

dissipation

alone

as

a

function

of

time,

Einstein

was

able to estimate the time interval below which his

results

are no

longer valid,

due to the breakdown

of

the assumed

independence

of

events in

successive time

intervals.[55]

V

Einstein's

further studies

of

fluctuation

phenomena

elaborate the two

fundamental

approaches

in his first two

papers on

Brownian motion: the

thermodynamical

approach

to

fluctuations that

emerged

from his work

on

statistical

mechanics,

and

the

stochastic treat-

ment

of

fluctuations that takes

dissipation

and time

dependence

into

account.[56]

By con-

centrating on a

few basic features

of

statistical

physics,

Einstein

further reduced

the need

for

detailed

microphysical assumptions

and

simplified

the technical

aspects

of

his

analysis.

All

of

these

subsequent

studies

emphasize applications

of

his

methods;

by

1909,

applica-

tions to

black-body

radiation

were again

of

primary concern

to

Einstein.[57]

Einstein further

developed

the

thermodynamical approach

in

a

paper

on voltage

fluctua-

tions in

a

condenser

(Einstein

1907b

[Doc. 39]).

He

gave a

simple

derivation of

a

formula

for

mean square fluctuations, a

derivation that does not

depend on dynamical premises,

but is

directly

based

on

Boltzmann's

principle relating probability

and

entropy, both

con-

ceived

by

Einstein

as thermodynamical quantities. Consequently,

he stated

that

his treat-

ment

of

fluctuations does not

require any "definite

stipulations

concerning

the

molecular

model

to

be

applied" ("bestimmte

Festsetzungen

in

betreff

des anzuwendenden moleku-

laren

Bildes").[58]

The

new

element in

Einstein's

argument

is

his definition

of

probability

in

Boltzmann's

principle as

the

"statistical

probability

of

a

state"

("statistische

Wahrscheinlichkeit eines Zustandes"),

a

concept

introduced

in

Einstein 1905i

(Doc.

14).[59]

Einstein further elaborated this

concept

in

Einstein 1909b

(Doc. 56),

where he

applied

the

mean square

fluctuation formula derived from

Boltzmann's

principle

to

energy

fluctuations in

black-body

radiation.[60]

[55]

For

a

discussion

of

subsequent

work

on

the

significance

of

such

a

time interval in the

analy-

sis

of

Brownian motion, see

Fürth

1922,

pp.

60-61, fn. 8.

[56]

For studies

of Einstein's

thermodynamical

approach

to

fluctuations, see

Klein

1967, 1974b,

1982a. For

a

review

of Einstein's

work on

the

stochastic

approach,

see

Sciama 1979.

[57]

See

Einstein

1909b

(Doc.

56);

for

a

dis-

cussion

of Einstein's

work

on black-body ra-

diation fluctuations,

see

the editorial

note,

"Einstein's

Early

Work

on

the

Quantum Hy-

pothesis," p.

146.

[58]

Einstein 1907b

(Doc.

39),

p.

569.

[59]

For

a

discussion

of the

concept

of

proba-

bility developed

in Doc. 14, see

the editorial

note,

"Einstein's

Early

Work

on

the

Quantum

Hypothesis,"

p.

138.

[60]

For further discussion

of this

paper,

see

the

editorial note,

"Einstein's

Early

Work

on

the

Quantum Hypothesis," pp.

145-146.