50 FOUNDATIONS
OF
STATISTICAL
PHYSICS
1902b
(Doc. 3)
is
the establishment
of
a
connection between
a
constant in
Einstein's
for-
mula
for
the canonical distribution
(defined below)
and
an
"observable
measure
of
tem-
perature"
("beobachtbare[s] Temperaturmaass")
(p. 425).
Following
the
programmatic
remark about the "gap" ("Lücke"),
Einstein
1902b
(Doc.
3)
defines
a very general
kind of
"mechanical
system"
("mechanisches
System"),
pos-
sessing a large
but finite number
of
degrees
of
freedom,
the
energy
of
which is
the
sum
of
a potential
term and
a
kinetic term that
is
quadratic
in the velocities. A
microcanonical
ensemble-a
virtual ensemble
of
N such
systems
with
energies
between E
and
E
+
5E—
is
introduced.
Using
Liouville's
theorem, and
assuming
that the
energy
is
the
only
explic-
itly time-independent
conserved
quantity
for
a
mechanical
system,
Einstein derived
an
equilibrium
distribution formula for this ensemble. A
more special
ensemble
satisfying
these
conditions
is
then
introduced,
consisting
of
coupled
systems S
and
S
(thermometer
and
heat
bath),
with the
energy
of
S
taken
to be
infinitely great compared
to that
of
S.
The
equilibrium
distribution
of
the
ensemble
of
subsystems
S
(a
canonical ensemble) is
derived:
dN
=
A"e'^dp1
...
dqn.
In the
course
of
this
derivation,
Einstein first introduced the structure
function
E + SE
o)(£)
bE
=
I
dp1
...
dpn,
J
E
which later
played an important
role in his
papers on
the
quantum hypothesis.[56]
After
establishing
the
above-mentioned connection between the constant h and
an
"ob-
servable
measure
of
temperature,"
used to
prove
the laws
of
thermal
equilibrium,
Einstein
derived the
equipartition
theorem for the canonical ensemble and the second law
of
thermodynamics
for reversible
processes, as
well
as an expression
for
the
entropy
of
a
mechanical
system
at
equilibrium,
e
=
E/T
+ 2
k
log{
Je~2hEdp1
...
dqn}
+ const.,
that
he
employed
several times in later
papers.[57]
Einstein
pointed
out that the
expression
for
the
entropy
is
noteworthy,
because it
depends solely on
E
and
T,
while
no longer allowing
the
special
form
[56]
See,
e.g.,
Einstein 1907a (Doc. 38), and
the editorial note,
"Einstein's
Early Work on
the Quantum
Hypothesis,"
p. 141. The structure
function
is
cited here
in the
form
first
given in Einstein 1904 (Doc. 5), p. 355, which
is the version that Einstein cited in his later
pa-
pers.
[57]
See,
e.g.,
Einstein 1905k (Doc. 16), p.
551,
and
Einstein
1906d (Doc. 34), p.
201.