48
FOUNDATIONS
OF
STATISTICAL PHYSICS
an
observable
measure
of
temperature
and the associated laws
of
thermal
equilibrium; (3)
his method
of
carrying
out the
derivation-he
independently
formulated the
distinction
between what
are
now,
following
Gibbs
1902,
called the microcanonical and the
canonical
ensembles; (4)
his
appreciation
of
certain crucial
conceptual problems
attendant
upon
the
introduction
of
probabilities
in
physics,
which led him to
assume
what would later be
called
an ergodic hypothesis,
in the form
of
the
equality
of
time and ensemble
averages;
and
(5)
his extraction
of
important new physical consequences
from his
foundational
stud-
ies,
including
most
notably
the
energy
fluctuation formula for the canonical
ensemble.
Einstein
1902b
(Doc.
3)
was
submitted to the Annalen in June 1902 and
published on
18
September.
It
opens
with the statement that Einstein intends to fill the
"gap"
("Lücke") in the mechanical foundations
of "the
general theory
of
heat" ("die
allge-
meine Wärmetheorie")
by deriving
the laws
of
thermal
equilibrium
and the second law
of
thermodynamics exclusively
from the
equations
of
mechanics and the
probability
calculus.
Einstein noted that
Maxwell's
and
Boltzmann's
theories
"have
already come near
to this
goal" ("diesem
Ziele bereits nahe
gekommen
sind"). In
a
letter written nine months
earlier,
he
gave a
similar
description
of
a
work that
may
have been
a
draft
of
part
of
this
paper:
Recently
I have
been
thoroughly
occupied
with
Boltzmann's
works
on
the
kinetic
theory
of
gases
&
have,
in
the
last few
days, myself written
a
small
paper,
which
provides
the
keystone
to
a
chain
of
derivations
begun by
him.
. . .
[I
will]
probably publish
it in the Annalen.
Ich habe mich in letzter
Zeit
gründlich
mit Boltzmanns Arbeiten über kine-
tische Gastheorie befaßt
&
in den letzten
Tagen
selbst eine
kleine
Arbeit
geschrieben,
welche den Schlußstein einer
von
ihm
begonnenen
Beweiskette
liefert.
. . .
[Ich werde]
sie wahrscheinlich in den Annalen
publizieren.[51]
It is not clear, however, just
what
shortcomings
Einstein
saw
in
Boltzmann's deriva-
tions. The
chief
novelties
of
Einstein 1902b
(Doc.
3) are
Einstein's
derivation
of
the
equi-
librium distribution for
a
virtual canonical ensemble
of
systems
from
the microcanonical
distribution, and his
subsequent employment
of
the canonical distribution to derive the
equipartition
theorem,
the laws
of
thermal
equilibrium,
and the second law for
systems
in
equilibrium.
This
suggests
that
Einstein
saw one shortcoming as
Boltzmann's
failure to
understand the
significance
of
the canonical
ensemble and its connection with the micro-
canonical ensemble.
Support
for this
hypothesis
comes
from
a
comment
by
Einstein in 1909
on
Planck's
[51]
Einstein to Marcel
Grossmann, 6 Septem-
ber
1901 (Vol.
1,
Doc.
122).
Parts
of
this earlier
paper may
survive in
§
8-10
of
Einstein 1902b
(Doc. 3).
These sections
provide
the
proof
of
a
thermodynamical assumption
crucial
to
the
ar-
gument
of
Einstein 1902a
(for
a
discussion
of
this
assumption, see
the editorial
note,
"Ein-
stein
on
the Nature
of Molecular Forces,"
p.
8).
And,
quite
independently
of
the rest
of
the
pa-
per, § 8
begins
with
a
reiteration
of
the charac-
terization
of
a
"mechanical
system"
("mechan-
isches
System") given
in
§
1.