50 FOUNDATIONS

OF

STATISTICAL

PHYSICS

1902b

(Doc. 3)

is

the establishment

of

a

connection between

a

constant in

Einstein's

for-

mula

for

the canonical distribution

(defined below)

and

an

"observable

measure

of

tem-

perature"

("beobachtbare[s] Temperaturmaass")

(p. 425).

Following

the

programmatic

remark about the "gap" ("Lücke"),

Einstein

1902b

(Doc.

3)

defines

a very general

kind of

"mechanical

system"

("mechanisches

System"),

pos-

sessing a large

but finite number

of

degrees

of

freedom,

the

energy

of

which is

the

sum

of

a potential

term and

a

kinetic term that

is

quadratic

in the velocities. A

microcanonical

ensemble-a

virtual ensemble

of

N such

systems

with

energies

between E

and

E

+

5E—

is

introduced.

Using

Liouville's

theorem, and

assuming

that the

energy

is

the

only

explic-

itly time-independent

conserved

quantity

for

a

mechanical

system,

Einstein derived

an

equilibrium

distribution formula for this ensemble. A

more special

ensemble

satisfying

these

conditions

is

then

introduced,

consisting

of

coupled

systems S

and

S

(thermometer

and

heat

bath),

with the

energy

of

S

taken

to be

infinitely great compared

to that

of

S.

The

equilibrium

distribution

of

the

ensemble

of

subsystems

S

(a

canonical ensemble) is

derived:

dN

=

A"e'^dp1

...

dqn.

In the

course

of

this

derivation,

Einstein first introduced the structure

function

E + SE

o)(£)

bE

=

I

dp1

...

dpn,

J

E

which later

played an important

role in his

papers on

the

quantum hypothesis.[56]

After

establishing

the

above-mentioned connection between the constant h and

an

"ob-

servable

measure

of

temperature,"

used to

prove

the laws

of

thermal

equilibrium,

Einstein

derived the

equipartition

theorem for the canonical ensemble and the second law

of

thermodynamics

for reversible

processes, as

well

as an expression

for

the

entropy

of

a

mechanical

system

at

equilibrium,

e

=

E/T

+ 2

k

log{

Je~2hEdp1

...

dqn}

+ const.,

that

he

employed

several times in later

papers.[57]

Einstein

pointed

out that the

expression

for

the

entropy

is

noteworthy,

because it

depends solely on

E

and

T,

while

no longer allowing

the

special

form

[56]

See,

e.g.,

Einstein 1907a (Doc. 38), and

the editorial note,

"Einstein's

Early Work on

the Quantum

Hypothesis,"

p. 141. The structure

function

is

cited here

in the

form

first

given in Einstein 1904 (Doc. 5), p. 355, which

is the version that Einstein cited in his later

pa-

pers.

[57]

See,

e.g.,

Einstein 1905k (Doc. 16), p.

551,

and

Einstein

1906d (Doc. 34), p.

201.