220
DOC.
8
ANALYSIS OF A RESONATOR'S MOTION
Doc. 8
Statistical
Investigation
of
a
Resonator's
Motion in
a
Field
by
A.
Einstein and
L.
Hopf
[Annalen
der
Physik
33
(1910):
1105-1115]
§1.
Train
of
Thought
It
has
been
shown in
a variety
of
ways
and
it
is
now generally
accepted
that,
when
correctly applied
in
the
theory
of
our
current views
on
the distribution
and
propagation
of
electromagnetic energy on
the
one
hand,
and
on
the
statistical
distribution
[1]
of
energy on
the other
hand,
can
to
no
other but the
so-called
Rayleigh
(Jeans)
law. Since this law
is
in
complete
with
experiment,
it
is
necessary
to
undertake
a
modification
of
the
foundations of
the
theories
used for its
derivation;
and
it has
often been
suspected
that the
application
of the
statistical
energy
distribution
laws
[2]
to
or
to
rapidly oscillating
motions
(resonators)
is not flawless.
The
following
investigation
shall
now
show
that
such
a
dubious
application
is not
required
at
all,
and
that
it suffices to
apply
the
equipartition
theorem for
energy solely
to
the
translational
motion of the
molecules and oscillators in
order
to
arrive at
the
Rayleigh
law.
The
applicability
of the
law to
translatory
motion
has
been
by
the
successes
of the kinetic
theory
of
gases;
we
may
therefore
conclude
that
only a
more
and
more
profound
change
in
our
fundamental
conceptions
can
to
a
law
of
[3]
is
in
better
agreement
with
experiment.
We consider
a
mobile
electromagnetic
oscillator1
that
is, on
the
one
hand,
subjected
to
the
effects
of
a
field
and, on
the other
hand,
possesses a
mass
m
and enters
into interaction
with
the
molecules
present
in
space.
If the
above
interaction
were
the
only one
present,
then the
mean
square
value
of the
momentum
associated with
the oscillator's
translatory
motion
would
be
completely
determined
by
statistical mechanics.
In
our case
there
also exists
the interaction of the oscillator
with
field.
For
a
statistical
equilibrium
to be
possible,
this
latter interaction
must
not
produce
any
change
in
that
mean
value.
In other
words:
The
mean
square
value
of
the
momentum associated with
the
translatory
motion that the oscillator
assumes
under
the influence of
be the
same as
that
which it would
assume,
in
accordance with
statistical
mechanics,
under the
mechanical influence
of the
molecules
alone. This
reduces the
problem
determining
the
mean
square
value
(mv)2
of the
momentum
assumed
by
the oscillator under the
sole influence
[4]
field.
This
mean
value must be
the
same
at time
t
=
0
as
at time
t
=
t, so
that
we
have
1
For the
sake
of
simplicity we
will
assume
that the
oscillator oscillates
only
in
the
z-direction and
moves only
in
the
x-direction.
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