74
DOC.
3
THEORY OF THERMAL
EQUILIBRIUM
1898a,
§
29, p. 82).
It
is
implicit
in Boltzmann
1871a, and
it
appears explicitly
in Boltzmann
1885,
pp.
78-79,
where Boltzmann first defined
the term
"Ergode" (see
note
6),
and
in Boltz-
mann
1898a,
§
32,
p.
89. The
striking
locution,
"unendlich
viele
(N) Systeme" ("infinitely
many
(N)
systems")
may
have been taken from
Boltzmann; exactly
the
same
words
are
used,
e.g.,
in Boltzmann
1887, p.
207.
[5]
Vi
denotes the internal
potential energy
of
the
system.
[6]
This condition
on
the
energy
is
implicit
in
Boltzmann 1871a,
p.
707, and its
significance
is
discussed
in Maxwell
1879, pp.
548-560.
It
is
employed
explicitly
in Boltzmann
1885,
pp.
78-
79, to define the
concept
of
an
"Ergode,"
a
vir-
tual
ensemble of
systems
with identical ener-
gies,
for which the
energy
is
the
only explicitly
time-independent
conserved
quantity.
The defi-
nition
of
an "Ergode"
in Boltzmann 1898a,
§
32,
p.
89,
does not include this condition.
[7]
The same
notation for constants
of
motion
is employed on pp.
78-79 of
Boltzmann
1885,
where the
concept
of
an
"Ergode" (see note 6)
was
first defined.
In Boltzmann 1898a the
same
notation is used,
more
generally,
to
represent
the
integrals
of
motion
of
a
system (see
§
30),
but it
is
not used in the discussion
of
an
"Ergode"
(§
32).
[8]
In Boltzmann 1898a,
§
25, uppercase
Pv
and
Qv
are
used
to
represent
coordinates and
mo-
menta
at
t
=
0,
with lowercase
pv
and
qv
representing
coordinates and momenta at
t
=
t;
G and
g are
used there to
designate
the associ-
ated
regions
of
phase space.
[9]
This
equation
should be
dN
= (px,
...
qn)
T
dpx
. . .
dqn.
J
8
[10]
Boltzmann
1898a.
[11]
The
idea
of
a
virtual ensemble
of
compos-
ite
systems
is
anticipated
in Boltzmann
1871a,
pp.
707-711, and in Boltzmann
1898a,
§
35,
where Boltzmann
considered
an
ensemble
of
systems,
each
of
which
is
divided into two
parts
separated
by
a
heat-conducting
wall.
[12]
The
symbol
H
represents
the
uppercase
Greek eta.
[13]
Cf. Boltzmann 1898a,
§
37,
p.
108, eq.
(115).
The constant h
was
first introduced
by
Boltzmann
in Boltzmann
1868,
p.
523,
and used
regularly
thereafter; see, e.g.,
Boltzmann
1896,
§
7, p.
48.
[14]
8E
should be 8E.
[15]
Such
a
distribution
is
now
called
a canon-
ical distribution,
following
Gibbs
1902,
pp.
33-
34. The
concept
of
such
an
ensemble
is
already
implicit
in Boltzmann
1872, pp.
368-370.
For
a
discussion
of Einstein's
later
adoption
of
Gibbs's
terminology,
see
the editorial
note,
"Einstein
on
the Foundations
of
Statistical
Physics,"
pp.
54-55.
[16]
Boltzmann referred
to
Kirchhoff
as
the
source
of
such
a probabilistic interpretation
of
this distribution
(see
Boltzmann 1898a,
§
38,
p.
112,
and
§
37).
Boltzmann
was
probably
refer-
ring
to
Kirchhoff 1894,
Lecture
13,
pp.
134-
141.
[17]
The limits
of
integration are
defined
by
((x')
taking a
value between
y'
and
y'
+
A,
where
y'
=
y/a2.
[18]
In
fact,
a
must
be
greater
than
1.
[19]
The
inequality
should
be
2
d[u(E1)] w(E2).
[20]
Einstein's
proof
of
what Paul Hertz
dubbed the
"Trennungssatz"
(the
assertion that
after
a composite system
Z
is
separated
into two
parts,
Z1
and
Z2,
h
=
h1
= h2)
is
strongly
criti-
cized in
Hertz,
P.
1910a, pp.
247-255.
Hertz
argued that,
in contrast to the
proof
of the
"Ver-
einigungssatz"
(the
assertion
that,
after two
sep-
arate
systems,
Z1
and
Z2, are
joined to
form
a
composite system
Z,
h1 = h2 =
h),
the
proof
of
the
"Trennungssatz"
requires
the
assumption
"that
all
phases
of
the
composite system are ex-
plored"
("daß alle Phasen des
zusammenge-
setzten
Systems
durchwandert werden")
(Hertz,
P. 1910a,
p.
254).
In his
reply
to
Hertz,
Einstein
wrote:
"The
...
objections
against
an
obser-
vation
on
thermal
equilibrium
contained in
my
first
essay
on
the
subject
rest
upon a
misunder-
standing
that
was brought
about
by an
all
too
terse
and
insufficiently
careful formulation of
that
observation"
("Die
...
Bemerkungen ge-
gen
eine in meiner
ersten
einschlägigen
Abhand-
lung
enthaltene
Betrachtung
über
das Tem-
peraturgleichgewicht
beruht
auf
einem
Mißverständnis,
das durch eine allzu
knappe
und nicht
genügend sorgfältige Formulierung je-
ner
Betrachtung hervorgerufen
wurde")
(Ein-
stein
1911c,
p.
175;
see
also
Einstein to Paul
Hertz,
14
August
1910).
Einstein did not
explain
the nature
of
the
misunderstanding.
For another
discussion
of
this
topic by
Einstein,
see
Einstein
1903
(Doc.
4),
§
4. See also Hertz, P.
1910b,
as
well
as
Ornstein
1910, 1911,
for
further discus–