DOC.
7
PROBABILITY CALCULUS
211
Doc.
7
On
a
Theorem
of
the
Probability
Calculus
and Its
Application
in
the
Theory
of
by
A.
Einstein and
L.
Hopf
[1]
[Annalen
der
Physik
33
(1910): 1096-1104]
§1.
The
Physical
Problem
as
the
Point
of
Departure
If
one
wants to
calculate
any
effect
in
the
theory
of
say
the
force
acting on
an
oscillator,
then
one
always
uses
Fourier
series
of the
general
form
sin
2m
-
+
B
cos
2m
-)
[2]
T "
T)
as
the
analytical expression
for the electric
or
the
magnetic
force.
The
problem is
here
immediately specialized
to
a
given spatial
point,
which
is
of
no importance
to what
follows;
t
denotes the
variable
time,
and
T
the
very long
time
period
for
which
the
series
applies.
When
calculating any average
values-and,
in
general,
only
such values
occur
in
the
theory
individual coefficients
An,
Bn
to be
independent
of
each
other,
one assumes
that each
coefficient follows
the
Gaussian
error
law
independently
of the numerical
values
of the other
coefficients, so
that the
probability1
dW of
a
combination of
values
An,
Bn
must
simply
be
the
product
of the
probabilities
of the
individual coefficients.
(1)
dW
=
WA.WA...WB.WB...dA..dB
Since
the
theory
of
in
the
form in which it follows
exactly
from
the
generally
accepted
foundations of
the
theory
of
electricity
and statistical mechanics
as we know,
to irresolvable
conflicts with
experience,
it
is
natural
to
mistrust
this
simple
assumption
of
independence
and to blame it
for the
failures
of the
theory
[4]
We
shall show in what follows
that
this
way
out
is
impossible,
and
that,
on
the
contrary,
the
physical
problem can
be
reduced
to
a purely
mathematical
problem
that
the
statistical law
(1).
That
is
to
say
that
if
we
consider
a
arriving
from
a
certain
direction,2
then
this
is
certain
to have
a
higher degree
of order than the total radiation
acting
at
a
point.
But
arriving
from
a
specific
direction
can
always
be conceived
as
arising
from
a
great
number of
emission
centers,
i.e.,
the
surface
that
emits
the
can
be
subdivided into
very many
surface
elements
that
1 By
"probability
of
a
coefficient"
we
obviously
have to
understand the
following:
We
imagine
that
the
electrical force
is
expanded
in
Fourier
series
for
very many
moments
of
time.
That fraction
of these
expansions
in which
a
coefficient lies
within
a specified range
of
values
is
the
probability
of
this
range
of
values
for the
coefficient considered. [3]
2
More
accurately:
"corresponding to
a
certain
elementary
angle
dk."
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