138 EARLY
WORK
ON QUANTUM
HYPOTHESIS
ergy
distribution
only
for small values
of
v/T; indeed,
as
Einstein
noted,
it
implies an
infinite total radiant
energy.[29]
Einstein
posed a question
in
1906,
which
preoccupied
him and others at the time:
"How
is it that Planck did not arrive at the
same
formula
[eq. (3)],
but at the
expression
...
[eq.
2]?" ("Woher
kommt
es,
daß Hr. Planck nicht
zu
der
gleichen
Formel,
sondern
zu
dem
Ausdruck
...
gelangt ist?")[30]
One
answer
lies
in
Planck's
definition
of
W
in Boltz-
mann's
principle, which,
as
Einstein
repeatedly
noted,
differs
fundamentally
from his
own
definition
of
probabilities as
time
averages.[31]
As noted
above,
Planck
interpreted W
as
proportional
to the number
of
complexions
of
a system.
As Einstein
pointed
out in
1909,
such
a
definition
of
W is
equivalent
to his definition
only
if
all
complexions
corresponding
to
a given
total
energy are equally probable.
However,
if
this
is
assumed to be
the
case
for
an
ensemble
of
oscillators in thermal
equilibrium
with
radiation,
the
Rayleigh-Jeans
law
results.[32] Hence,
the
validity
of Planck's
law
implies
that all
complexions
cannot be
equally probable.
Einstein showed
that,
if
the
energies
available to
a
canonical
ensemble
of
oscillators
are arbitrarily
restricted to
multiples
of
the
energy
element
hv,
then all
pos-
sible
complexions are
not
equally probable,
and
Planck's
law
results.[33]
A
third
element
of Einstein's
work
on
statistical
physics
that
is
central to his
work
on
the
quantum
hypothesis
is
his method for
calculating
mean
square
fluctuations in the state
variables
of
a system
in
thermal
equilibrium.
He
employed
the canonical
ensemble
to
calculate such fluctuations
in
the
energy
of
mechanical
systems,
and then
applied
the
result
to
a
nonmechanical
system-black-body radiation,
deducing a
relation
that
agrees
with
Wien's
displacement
law.[34]
This
agreement
confirms the
applicability
of
statistical
con-
cepts
to radiation, and
may
have
suggested
to
him the
possibility
that radiation could be
treated
as a system
with
a
finite number
of
degrees
of
freedom,
a
possibility
he
raised
at
the outset
of
his first
paper on
the
quantum hypothesis.[35]
In connection with his work
on
Brownian motion
in
1905-1906, Einstein
developed
additional methods
for
calculating
fluctuations,
methods which
he
later
applied
to the anal-
ysis
of
black-body
radiation. In
particular,
he
developed a
method based
on
the inversion
of
Boltzmann's
principle,
which
may
be used
even
in the absence
of
a microscopic
model
[29]
See Einstein 1905i
(Doc. 14), p.
136.
Ehrenfest
later called
this
divergent
behavior the
"Rayleigh-Jeans
catastrophe
in the
ultraviolet"
("Rayleigh-Jeans-Katastrophe
im Ultraviolet-
ten")
(Ehrenfest 1911,
p.
92).
[30]
Einstein
1906d
(Doc. 34), p.
200.
The
problem
is also discussed
in
Einstein 1907a
(Doc. 38)
and Einstein 1909b
(Doc. 56); and,
e.g.,
in
Ehrenfest
1906 and
Rayleigh
1905b.
[31]
The difference between their definitions
is
stated
particularly clearly
in Einstein 1909b
(Doc.
56),
sec.
4,
pp.
187-188. Einstein first
gave
his definition in Einstein 1903
(Doc.
4),
pp.
171-172. Einstein's
definition is discussed
in the editorial
note,
"Einstein
on
the Founda-
tions
of
Statistical
Physics,"
p.
52;
Klein 1974b
and
Pais
1982,
chap.
4.
[32]
See
Einstein
1909b
(Doc. 56), p.
187.
[33]
See Einstein 1906d
(Doc. 34),
pp.
201-
203,
and
Einstein 1907a
(Doc. 38),
pp.
182-
184.
[34]
See
Einstein
1904
(Doc. 5),
especially p.
362. For further discussion
of
his work
on en-
ergy
fluctuations, see
Klein
1967,
and the edi-
torial
notes,
"Einstein
on
the Foundations
of
Statistical
Physics,"
p.
54,
and
"Einstein
on
Brownian Motion,"
pp.
206-222.
[35]
See Einstein 1905i
(Doc. 14), pp.
132-
133.
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