138 EARLY

WORK

ON QUANTUM

HYPOTHESIS

ergy

distribution

only

for small values

of

v/T; indeed,

as

Einstein

noted,

it

implies an

infinite total radiant

energy.[29]

Einstein

posed a question

in

1906,

which

preoccupied

him and others at the time:

"How

is it that Planck did not arrive at the

same

formula

[eq. (3)],

but at the

expression

...

[eq.

2]?" ("Woher

kommt

es,

daß Hr. Planck nicht

zu

der

gleichen

Formel,

sondern

zu

dem

Ausdruck

...

gelangt ist?")[30]

One

answer

lies

in

Planck's

definition

of

W

in Boltz-

mann's

principle, which,

as

Einstein

repeatedly

noted,

differs

fundamentally

from his

own

definition

of

probabilities as

time

averages.[31]

As noted

above,

Planck

interpreted W

as

proportional

to the number

of

complexions

of

a system.

As Einstein

pointed

out in

1909,

such

a

definition

of

W is

equivalent

to his definition

only

if

all

complexions

corresponding

to

a given

total

energy are equally probable.

However,

if

this

is

assumed to be

the

case

for

an

ensemble

of

oscillators in thermal

equilibrium

with

radiation,

the

Rayleigh-Jeans

law

results.[32] Hence,

the

validity

of Planck's

law

implies

that all

complexions

cannot be

equally probable.

Einstein showed

that,

if

the

energies

available to

a

canonical

ensemble

of

oscillators

are arbitrarily

restricted to

multiples

of

the

energy

element

hv,

then all

pos-

sible

complexions are

not

equally probable,

and

Planck's

law

results.[33]

A

third

element

of Einstein's

work

on

statistical

physics

that

is

central to his

work

on

the

quantum

hypothesis

is

his method for

calculating

mean

square

fluctuations in the state

variables

of

a system

in

thermal

equilibrium.

He

employed

the canonical

ensemble

to

calculate such fluctuations

in

the

energy

of

mechanical

systems,

and then

applied

the

result

to

a

nonmechanical

system-black-body radiation,

deducing a

relation

that

agrees

with

Wien's

displacement

law.[34]

This

agreement

confirms the

applicability

of

statistical

con-

cepts

to radiation, and

may

have

suggested

to

him the

possibility

that radiation could be

treated

as a system

with

a

finite number

of

degrees

of

freedom,

a

possibility

he

raised

at

the outset

of

his first

paper on

the

quantum hypothesis.[35]

In connection with his work

on

Brownian motion

in

1905-1906, Einstein

developed

additional methods

for

calculating

fluctuations,

methods which

he

later

applied

to the anal-

ysis

of

black-body

radiation. In

particular,

he

developed a

method based

on

the inversion

of

Boltzmann's

principle,

which

may

be used

even

in the absence

of

a microscopic

model

[29]

See Einstein 1905i

(Doc. 14), p.

136.

Ehrenfest

later called

this

divergent

behavior the

"Rayleigh-Jeans

catastrophe

in the

ultraviolet"

("Rayleigh-Jeans-Katastrophe

im Ultraviolet-

ten")

(Ehrenfest 1911,

p.

92).

[30]

Einstein

1906d

(Doc. 34), p.

200.

The

problem

is also discussed

in

Einstein 1907a

(Doc. 38)

and Einstein 1909b

(Doc. 56); and,

e.g.,

in

Ehrenfest

1906 and

Rayleigh

1905b.

[31]

The difference between their definitions

is

stated

particularly clearly

in Einstein 1909b

(Doc.

56),

sec.

4,

pp.

187-188. Einstein first

gave

his definition in Einstein 1903

(Doc.

4),

pp.

171-172. Einstein's

definition is discussed

in the editorial

note,

"Einstein

on

the Founda-

tions

of

Statistical

Physics,"

p.

52;

Klein 1974b

and

Pais

1982,

chap.

4.

[32]

See

Einstein

1909b

(Doc. 56), p.

187.

[33]

See Einstein 1906d

(Doc. 34),

pp.

201-

203,

and

Einstein 1907a

(Doc. 38),

pp.

182-

184.

[34]

See

Einstein

1904

(Doc. 5),

especially p.

362. For further discussion

of

his work

on en-

ergy

fluctuations, see

Klein

1967,

and the edi-

torial

notes,

"Einstein

on

the Foundations

of

Statistical

Physics,"

p.

54,

and

"Einstein

on

Brownian Motion,"

pp.

206-222.

[35]

See Einstein 1905i

(Doc. 14), pp.

132-

133.