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.