268
DOC.
7
PROBABILITY CALCULUS
Published in Annalen
der
Physik 33
(1910):
1096-1104.
Received 29
August 1910, published
20
December 1910.
[1]Ludwig
Hopf
(1884-1939),
who had received
his
degree
with Sommerfeld in
1909, was
registered
in all
three of Einstein's
courses
at
the
University
of Zurich
in
the
summer
semester
1910.
He collaborated with Einstein
not
only on
this
paper
and the
subsequent
one (Doc.
8),
but also
helped
him
to identify
a
calculational
error
in
an
earlier
publication;
see
Einstein
1911e
(Doc.
14)
and the
notes to
this document. For evidence of their
personal relationship
at
the
time of their collaboration
on
the
present paper, see
also Einstein
to Ludwig
Hopf, 21
June
1910,
and Einstein
to Ludwig
Hopf,
2
August
1910.
[2]For the
use
of
a
Fourier
decomposition
of
the radiation
field in
the
study
of heat
radiation,
see,
e.g.,
the classic book
by
Planck,
Planck
1906,
in
particular pp. 118ff.
Einstein had earlier
reviewed this
book;
see
Einstein
1906f (Vol.
2,
Doc.
37).
He
quoted
it
in his
second
paper
with
Hopf;
see
Einstein and
Hopf
1910b (Doc.
8), p.
1107, fn. 2.
[3]This
definition of
probability is
different from
Planck's;
for
Planck's
definition,
see
Planck
1906,
p. 139;
for
a
historical discussion of the relevance of
this
difference, see
the editorial
note
in Vol.
2,
"Einstein's
Early
Work
on
the
Quantum
Hypothesis,"
pp. 137-139,
and Kuhn
1978,
in
particular pp.
182-187.
[4]Contemporary
applications
of the
equipartition
theorem
to
heat radiation
assumed–
either
explicitly
or
implicitly-the
statistical
independence
of the
Fourier
coefficients and
led to
the
experimentally
refuted
Rayleigh-Jeans
law;
for historical
accounts,
see
Klein,
M.
1970,
pp.
234-237,
and Kuhn
1978,
pp.
143-152.
This
assumption
also
corresponds to
Planck's
hypothesis
of "natural radiation"
("natürliche Strahlung");
for
its
description
by
Planck
in
terms
of the Fourier
coefficients, see, e.g.,
Planck
1906, p. 133,
and
for its
role
in Planck's
analysis
of
black-body radiation,
see
pp.
187ff.
For
Einstein's views
on
the statistical
properties
of radiation
as
distinct
from those
of Planck in
1910, see
Einstein's
unpublished response to
Planck 1910a
(Doc.
3).
[5]The
inference
expressed in
the last
sentence
later
gave rise
to
a
controversy
between
Einstein and Max
von
Laue;
see
Laue
1915a,
Einstein
1915b,
and Laue
1915b.
In
1924
Planck
considered the
question
as
still
unresolved;
see
Planck
1924.
This discussion
is
mentioned
in
Klein,
M.
1964, p. 16.
[6]The
mathematical
problem
treated
by
Einstein and
Hopf belongs
to
the tradition of
cen-
tral
limit
theorems, going
back
at
least
to
Laplace
(see, e.g.,
Stigler 1986, pp.
136ff)
and
actively
pursued
around the
turn
of
this
century,
in
particular
by
the Russian school
(see, e.g.,
Maistrov
1974,
pp.
208ff).
At
the time of the
present
paper, however, probability theory
was
not yet
a
mathematical
discipline
with
a
standard literature to which
physicists
would
commonly
refer
(see,
for
instance,
the
complaint
about
the
neglect
by
physicists
of results achieved
in
statistics
in Ehrenfest and Ehrenfest
1911,
pp. 86-87).
In the criticism of Einstein's and
Hopf's
result
mentioned in note
3,
however,
von
Laue did
refer to
the
recently published
German translation
of the
Russian textbook
on
probability theory
by
Markoff,
Markoff
1912
(see
Laue
1915a,
p. 855,
fn.
1).
[7]The
following argument
derives
a
differential
equation
for
a
diffusion
process
in
a
way
similar
to
the
one
Einstein had used earlier in his
analysis
of Brownian
motion; see
Einstein
1905k
(Vol.
2,
Doc.
16),
pp.
557-558.
[8]There
should be
a
minus
sign
in front
of
the
right-hand
side of
the
equation.
[9]The
right-hand
side
of
this
equation
should
be
continued
by
"...,"
since only
the first
two
factors
are
written
out.
[10]The
an
of the
right-hand
side
should
be
an.