APPENDIX

A 585

derived

(a

similar discussion

occurs

in

Einstein’s

lecture notes

for

a course on

kinetic

theory

in 1910

[see

Vol.

3,

Doc.

4,

pp.

234-235]).

This

is next

applied

to

osmotic

pressure

and diffusion.

Finally,

two

topics

of fundamental

importance

are

discussed: the derivation of

the

second law of

thermodynamics

from statistical

mechanics,

and Boltzmann’s

princi-

ple.

The second law

is

derived from the invariance of the

norm

of the distribution

function for small variations of the

temperature

and of the

parameters determining

the

energy.

This invariance is shown

to

imply

that

dQ/T

is

a

total differential. An

ex-

pression

for the

entropy

is

derived

as

well. Boltzmann’s

principle

and the relation be-

tween

irreversibility

and

probability

are

discussed in the

final

lecture. The discussion

is illustrated with

an

example

that Einstein had also used

on

earlier occasions

(see,

in

particular,

Einstein

et

al.

1914,

pp.

355-356):

a

fluid

containing

a

particle

whose

density

is

greater

than that of the

surrounding

fluid.

Although

the

particle

will fall

most

of

the time,

because lower

positions

have

higher

probability,

it

will also

some-

times

move

upward,

thus

illustrating

the statistical

nature

of

irreversibility.