244

DOC.

4

KINETIC THEORY LECTURE NOTES

drift

component along

the

tube,

which

he

denotes

by u.

Taking

the

momentum

transferred to

the

wall to be

equal

to

mu

presupposes

that the

walls reflect

the

colliding

molecules

completely

diffusively.

This

is

a

crucial

assumption

discussed

at

some length

by

Knudsen;

see,

e.g.,

Knudsen

1909a,

pp.

77,

104-105. Einstein's derivation

is

simplified

in

comparison

with Knudsen's

dis-

cussion,

in

that

the Maxwellian

velocity

distribution of

the

molecules

is not

taken into

account.

[32]P denotes the circumference of the tube.

[33]See

Knudsen

1909a.

[34]Here

Einstein

is following

Knudsen

1910a.

Further

experimental

evidence

was

given

in

Knudsen

1910b.

[35]See

Brush

1976,

§5.5,

for

a

historical discussion

concerning

the radiometer.

[36]For

a

discussion of Einstein's

own

earlier work

on

statistical

physics,

see

Vol.

2,

the

editorial

note,

"Einstein

on

the Foundations of Statistical

Physics," pp.

41-55.

[37]The

distinction

among

several kinds of

trajectories

that

is

made here

was

made earlier

by

Gibbs

(see

Gibbs

1905, chap.

12).

Einstein's three

cases are

related

to

the distinction

among

what became known later

as periodic, nonergodic,

and

quasiergodic

trajectories. See Ehrenfest

and

Ehrenfest

1911

for

a contemporary

discussion and Bernhardt

1971,

Brush

1976,

and Plato

1991

for further

references.

[38]The

spiral

of Archimedes

is

the

curve

traced

by

a

point moving uniformly along

a

straight

line

while

at

the

same

time

rotating

uniformly

around

a

fixed

point on

the

same

line.

See,

e.g.,

Loria

1902

for details.

[39]The

relation

lim x/T

=

lim n/N is

also derived

on

a

loose

sheet,

available

only in photo-

copy,

inserted

at

the

beginning

of

the

notebook and

presented

as

[p.

55].

For

earlier

treatments

of this

question see,

e.g.,

Einstein

1903

(Vol.

2,

Doc.

4), §2,

and the discussion in

Vol.

2,

the

editorial

note,

"Einstein

on

the Foundations of Statistical

Physics,"

p.

52.

[40]The

following example

of

a point

moving

on a

torus is

also discussed in

Ehrenfest and

Ehrenfest 1911,

pp. 31-32, fn.

89a.

[41]A

similar

expression ("unendlich viele

(N)

Systeme") is

used

in

Einstein

1902b

(vol. 2,

Doc.

3),

p.

58.

[42]For

an

extensive discussion

of

Liouville's theorem

(of

which this result

is

a

version),

see

Boltzmann

1898, §§25-29.

Gibbs discussed

Liouville's theorem under the title of "conservation

of

density-in-phase" (Gibbs 1902,

pp. 9-11)

and

"Erhaltung

der Phasendichte"

(Gibbs 1905,

p. 8),

respectively.

For

contemporary

discussions

see,

e.g.,

Ehrenfest

and

Ehrenfest 1911, pp.

27-

29;

Wassmuth

1915,

pp.

4-7; and Hertz, P.

1916,

pp.

455-460.

[43]See

[p.

16].

[44]The

same general approach

in

which

no

distinction

was

made between coordinates and

momenta

was

taken

in

Einstein

1903

(Vol.

2,

Doc.

4), §1.

[45]On

[p. 19]

Einstein

uses

the term

"Bewegungsgesetz," on [p. 23]

he

uses

the

term

"Verän-

derungsgleichungen."

In Einstein 1903

(Vol. 2,

Doc.

4), §1,

the

more general

term

"Verände-

rung"

is

employed.

[46]See

Gibbs

1905, chap.

10, p.

117.

[47]"Kanonische Gesamtheit"

is

corrected from "Kanonisches

Gesamtsystem."

Einstein

wavered between these

terms,

as

is

clear from similar corrections later

on

and

some

in-

stances

where the

term

"Gesamtsystem"

is

used,

but the

term

"Gesamtheit" would be

more

appropriate.

[48]The

same assumption

is

made

in

Einstein

1902b

(Vol.

2,

Doc.

3),

p. 418,

and

in

Einstein

1903

(Vol. 2,

Doc.

4),

pp.

170-171.

[49]The constant ©

was

called "Modul" of the distribution

in Gibbs

1905,

chap.

4, p.

32.

In

his early papers

on

statistical

physics,

Einstein had

employed

the notation

2h

instead of

1/©;

see

Einstein

1902b

(Vol.

2,

Doc.

3),

and Einstein

1903

(Vol.

2,

Doc.

4).

The notation used here

is

adopted

from

Gibbs

1902, 1905.

See

also

notes 67

and

72.

[50](^

]

=

0, etc., as a

result of Hamilton's

equations

of

motion,

see [p. 37]

for

dx1 m

yoXi

a

related

argument.

[51]See note 41.