BROWNIAN MOTION 213

notions

of

semipermeable

membrane and osmotic

pressure

in

his

correspondence

with

Michele Besso,

showing

interest in

Sutherland's

hypothesis

on

the mechanism

of

semi-

permeable

membranes.[47]

In his

papers

on

statistical

physics,

Einstein

generalized

the

idea

of

external

conservative

forces,[48]

and noted the

significant

role

of

fluctuations in

statistical

physics.

In Einstein 1904

(Doc.

5)

he derived

an expression

for

mean square

deviations from the

average

value

of

the

energy

of

a

system.[49]

His second

paper on

Brownian

motion,

Einstein 1906b

(Doc. 32),

shows

"how

Brownian motion

is

related

to

the foundations

of

the molecular

theory

of heat" ("wie

die

Brownsche

Bewegung

mit den

Grundlagen

der

molekularen Theorie der Wärme

zusammenhängt").[50]

It includes two

new

fluctuation

formulas,

both

of

which

are

derived

from

the

probability

distribution for

a

canonical ensemble

given

in

Einstein's

papers on

statistical

physics.[51]

The first formula

(eq.

[I]

on

his

p. 373),

which

is

closely

related to

the formula for

energy

fluctuations he had derived in

1904,[52] gives

the

probability

of

deviations from the

equilibrium

value,

due

to

irregular

molecular motions,

of

a

suitable

observable

parameter

a

of

a system subject

to

an

external force with

potential

&(a):[53]

dW

=

A'

e'NRTda,

(I)

where dW

is

the

probability

that the value

of the

parameter

lies

between

a

and

a

+ da,

and A'

a

constant. Einstein

applied eq.

(I)

to

a

harmonic oscillator in

equilibrium

with

a

gas

to derive the

black-body

radiation law in the limit

of

large wavelengths

and

high

tem-

peratures.

An

investigation

of

how small

a particle

must be

in order

to remain

in

suspen-

sion in

a gravitational

field

provides

another

application.

Eq. (I)

does

not, however,

allow the treatment

of

Brownian

motion,

a

time-dependent

process involving

the

interplay

of

fluctuations and

dissipation.

In order to derive

a

fluctua-

tion formula that

generalizes eq.

(1),

Einstein related

a general dissipation

mechanism,

analogous

to

Stokes's

law,

to the condition for

stability

of

the distribution,

eq.

(I),

now

interpreted as giving

the number

of

systems

in

a

certain

state,

rather than the

probability

of

that state. The

potential

I

refers to

a

fictitious

force.[54]

The

resulting

formula for

the

time

dependence

of

the

mean square

fluctuation

(eq.

[II] on

his

p.

378)

enabled Einstein

to treat rotational

as

well

as

translational

motions

of

suspended particles:

[47]

See

Michele

Besso to Einstein,

7-11 Feb-

ruary

1903,

which indicates that there

was

ad-

ditional

correspondence on

this

subject.

See

Sutherland

1897 for

Sutherland's

hypothesis.

[48] See,

in

particular,

Einstein 1902b

(Doc.

3),

§

10.

[49]

At this

time, however,

Einstein

regarded

black-body

radiation

as

the

only physical system

for which

experience suggests

the existence of

observable

energy

fluctuations

(see

Einstein

1904

[Doc.

5],

p.

361).

[50]

Einstein

1906b

(Doc. 32),

p.

371.

[51]

See Einstein 1902b

(Doc. 3),

§

3.

[52]

See Einstein 1904

(Doc. 5),

§

4.

[53]

The choice

of

the observable

parameter,

which

is

treated

by analogy to

the

energy

in

the

fluctuation formula

given

in Einstein 1904

(Doc.

5),

was

the

subject

of

correspondence

between

D.K.C.

MacDonald and Einstein in 1953.

[54]

For

a

discussion

of Einstein's

use

of

such

fictitious forces,

see

the editorial

note,

"Ein-

stein's

Dissertation

on

the

Determination

of

Mo-

lecular Dimensions,"

§

IV,

p.

177.