FOUNDATIONS
OF
STATISTICAL PHYSICS
49
attempt
to
deny
the
applicability
of
the
equipartition
theorem
to
radiation.[52] Planck,
citing
Boltzmann,
argued
that
a necessary
condition for the
validity
of
the
equipartition
theorem
is
that the state distribution be
an ergodic
one,
in the
sense
that the
probability
of
a sys-
tem's
occupying a given
cell
of
phase space
is
proportional
to the size
of
the cell,
however
small the cell
may
be;
but that
an ergodic
distribution is not
guaranteed
if
the
phase space
is
partitioned
into cells
of
finite
size,
as
in
Planck's
version
of
the
quantum
hypothesis.[53]
Einstein wrote:
The
attempt
...
to call into
question
the
general validity
of
equation
II
[the
equipartition
theorem]
rests-it
seems
to me-solely
on a gap
in
Boltzmann's
inquiries,
which has been filled
in
the meantime
by
Gibbs's
investigations.
Der
...
Versuch,
die
Allgemeingültigkeit
der
Gleichung
II in
Frage zu
stel-
len,
beruht-wie
mir
scheint-nur
auf einer Lücke
in
Boltzmanns Betrach-
tungen,
welche unterdessen durch die Gibbsschen
Untersuchungen ausgefüllt
wurde.[54]
Gibbs, like
Einstein,
showed that the
equipartition
theorem holds
not
only
for
an
ergodic
or
microcanonical ensemble
(which
is
what Planck
apparently
took Boltzmann to have
asserted),
but also for
a
canonical
ensemble.[55]
Einstein
may
have considered another
shortcoming
of Boltzmann's
approach
to be that,
as a
consequence
of
his failure
to
grasp
the
significance
of
the canonical ensemble, he
could not construct
a
link between the fundamental
concepts
of
the kinetic
theory
and the
measurement
of
crucial
macroscopic thermodynamical quantities
like
temperature.
Such
measurements
require physical
interactions
of
a
kind that
can only
be treated
theoretically
with the introduction
of
the canonical ensemble. In fact,
one
of
the main
topics
of
Einstein
[52]
See Einstein 1909b
(Doc. 56),
pp.
186-
187. This comment
was
made in the context
of
Einstein's
criticism
of Planck's
derivation
of
the
radiation
law.
For
a
discussion
of
this
criticism,
see
the
editorial note,
"Einstein's
Early
Work
on
the
Quantum Hypothesis,"
p.
138.
[53]
See
Planck
1906c,
p.
178. Planck
cites
Boltzmann
1898a,
p. 101,
where
one
reads:
"Of
course
this
equality
of
the
mean
value
of
the ki-
netic
energy corresponding
to
each momentoid
has
only
been
proved
for the assumed
(ergodic)
state distribution.
. . .
But in
general
there
can
and will be other
stationary
state distributions,
for which these theorems
are
not
valid"
("Selbstverständlich ist diese
Gleichheit des
Mittelwerthes der
jedem Momentoide
entspre-
chenden
lebendigen
Kraft
nur
für
die
vorausge-
setzte
(ergodische) Zustandsvertheilung
bewie-
sen.
. . .
Es kann
und wird im
Allgemeinen
aber
auch
andere stationäre
Zustandsvertheilungen
geben,
für welche diese Sätze nicht
gelten").
In
Boltzmann
1898a,
§
32,
and
earlier
in Boltz-
mann
1885,
an
"Ergode"
is
defined
as a
virtual
ensemble
of
systems
with identical
energies,
where the
energy
is
the
only time-independent
conserved
quantity.
"Momentoide"
is
Boltz-
mann's
term
for
"degrees
of
freedom";
see
Boltzmann 1898a,
§
33. For
a
discussion
of
Planck's
argument, see
the
editorial
note,
"Ein-
stein's
Early
Work
on
the
Quantum Hypothe-
sis,"
p.
138.
[54]
Einstein
1909b
(Doc. 56),
p.
186;
see
also
Einstein and
Hopf
1910b.
[55]
See
Einstein
1902b
(Doc. 3),
pp.
427-
428,
and Gibbs
1902,
p.
49, which
corresponds
to
Gibbs
1905,
p.
48. A few
pages
after the
pas-
sage
cited
by
Planck,
Boltzmann
himself
proved
the
equipartition theorem for
a
real
ensemble
of
canonically distributed
systems
whose interac-
tions
are
assumed
to
be
negligible
(see
Boltz-
mann
1898a,
§
42),
but nowhere did he
prove
it
for
a
canonically
distributed virtual ensemble.