DOC.
4
FOUNDATIONS OF
THERMODYNAMICS 95
Published in Annalen
der
Physik 11
(1903):
170-187.
Dated
Bern,
January
1903,
received
26
January
1903,
published
16
April
1903.
[1]
Einstein
1902b
(Doc. 3).
[2]
It
is not
entirely
clear what Einstein meant
by
the
"kinetische
Theorie der
Wärme"
("ki-
netic
theory
of
heat");
consequently,
it is not
clear
exactly
what kind
of
generalization
Ein-
stein
was aiming
for here. In the cited
paper
Ein-
stein
described his aim
as being
"to
derive the
laws
of
thermal
equilibrium
and the second law
by using exclusively
the
equations
of
mechanics
and
the
probability
calculus"
("die
Sätze über
das
Wärmegleichgewicht
und
den
zweiten
Hauptsatz
unter
alleiniger
Benutzung
der
me-
chanischen
Gleichungen
und
der Wahrschein-
lichkeitsrechnung
herzuleiten")
(p.
417).
The
principal
difference between that
paper
and the
present one
is
that here
no
distinction
is
made
between coordinates and
momenta
in the defini-
tion
of
what Einstein called
a
"physical system"
("physikalisches System")
(p.
170),
rather than
a
"mechanical
system"
("mechanisches
Sys-
tem")
(Einstein
1902b
[Doc. 3], p. 417).
In
par-
ticular, therefore,
there
is
no
distinction here be-
tween
kinetic and
potential energy;
and
avoidance
of
this distinction
was precisely
the
kind
of
generalization
entertained
near
the end
of
the
previous paper
(see
Einstein 1902b
[Doc.
3], p. 433).
[3]
The
state variables,
p1, p2,
...,
Pn,
here
replace
both the coordinates
and
momenta
used
in
Einstein
1902b.
Boltzmann
1871a, p. 679,
employs a
similar
set
of
variables
to characterize
the state
of
a system.
The
temporal
evolution of
these variables is
governed by a
set
of
first-order
differential
equations
similar
to eqs. (1)
in
the
text.
[4]
This
is
the
same assumption
made in Ein-
stein 1902b
(Doc. 3),
p.
418
(see
especially
note
6).
In
a
letter
to
Michele Besso
of
17
March
1903,
Einstein wrote:
"The
condition that
E
be
the
only integral
of
the
equations
of
the
given
form is
no
limitation,
since
I
free
myself
of
it
when
considering
'adiabatically'
influenced
sys-
tems"
("Die
Bedingung,
daß
E
das
einzige
In-
tegral
der
Gleichungen
sei
von
der bewußten
Form ist keine
Einschränkung,
weil ich mich
bei
Betrachtung
der sich
'adiabatisch'
beeinflussen-
den
Systeme
davon befreie"). The
concept
of
adiabatic influence
is
defined
below,
on
p.
179.
L. S. Ornstein
questioned
whether
Einstein's
as-
sumption
is sufficient
to guarantee
that the
sys-
tem
would
explore
with
equal frequency
all
of
the
phase points lying on
the
hypersurface
of
phase space
defined
by
the energy's
being
the
only
conserved
quantity;
see
Ornstein
1910, pp.
805-808.
[5]
The definition
of
probabilities by
time
av-
erages,
and the
equating
of
time and ensemble
averages, is
implicit
in Boltzmann
1868,
pp.
517-518, Boltzmann
1871a,
pp.
691, 708,
and
Boltzmann 1898a,
§
35,
p.
103.
Einstein's
ex-
plicit equating
of
time and ensemble
averages
is
foreshadowed in Einstein 1902b
(Doc.
3),
p.
430
(see
note
34).
It
was
later criticized
by
Orn-
stein,
who
objected
that "[i]t
is
impossible
for
a
system
to
pass rigidly
in
a
finite time
(whatever
length
it
may
have)
through
all the
possible
phases;
and in
calculating a time-average
such
a
finite time must
be
assumed"
(Ornstein
1910,
p.
805).
Ornstein's
criticism
of
the
equating
of
time and
ensemble
averages
is
related
to
his crit-
icism
of
the
proofs given
in
Hertz,
P. 1910a
of
the
"Vereinigungssatz"
and the "Trennungs-
satz," theorems discussed
by
Einstein here
(§
4)
and
earlier
in Einstein 1902b
(Doc. 3),
§
5 (see
especially
note 20).
There
is
no
record
of
a reply
by
Einstein
to Ornstein; see, however,
Hertz's
reply,
Hertz, P. 1910b. See also Lorentz 1916.
[6] Presumably
in
response
to
a question
from
Besso, Einstein
explained
his
reasoning
in this
part
of
the
paper
in his letter to Besso
of
17
March 1903:
"First of
all about the
EgeQv,
dpv.
If
one interprets
p1
...
pn
as
coordinates in
an
n-dimensional
space,
then the
system
corre-
sponds
to
a point. e
is
the
density
of
points,
ecpv
are
the
components
of
the
material
flow,
and the
above
expression
the solenoidal
condition"
("Zuerst
zu
der
E
a€(Pv/dpv
Interpretiere
P1
...
pn
als Koordinaten in einem n-dimensionalen
Raume, dann
entspricht
dem
Systeme
ein Punkt.
E
ist die
Dichte der
Punkte, €v
sind die
Kom-
ponenten
der materiellen
Strömung,
und der
obige
Ausdruck die Solenoidalitätsbedin-
gung").
The first
equation on p.
173 is
a
form
of
the
equation
of
continuity
for the
points.
[7]
The
validity
of
the
preceding argument re-
quires
that
X
-
=
0
(the incompressibility
dpv
condition
for the
points
of the
state
space),
since
de/dt must vanish for
a stationary
distribution.
Einstein
evidently
had second
thoughts
about
the
argument
even
before the
paper
was published.
In the
previously
cited letter
to
Besso
of
17