BROWNIAN MOTION 213
notions
of
semipermeable
membrane and osmotic
pressure
in
his
correspondence
with
Michele Besso,
showing
interest in
Sutherland's
hypothesis
on
the mechanism
of
semi-
permeable
membranes.[47]
In his
papers
on
statistical
physics,
Einstein
generalized
the
idea
of
external
conservative
forces,[48]
and noted the
significant
role
of
fluctuations in
statistical
physics.
In Einstein 1904
(Doc.
5)
he derived
an expression
for
mean square
deviations from the
average
value
of
the
energy
of
a
system.[49]
His second
paper on
Brownian
motion,
Einstein 1906b
(Doc. 32),
shows
"how
Brownian motion
is
related
to
the foundations
of
the molecular
theory
of heat" ("wie
die
Brownsche
Bewegung
mit den
Grundlagen
der
molekularen Theorie der Wärme
zusammenhängt").[50]
It includes two
new
fluctuation
formulas,
both
of
which
are
derived
from
the
probability
distribution for
a
canonical ensemble
given
in
Einstein's
papers on
statistical
physics.[51]
The first formula
(eq.
[I]
on
his
p. 373),
which
is
closely
related to
the formula for
energy
fluctuations he had derived in
1904,[52] gives
the
probability
of
deviations from the
equilibrium
value,
due
to
irregular
molecular motions,
of
a
suitable
observable
parameter
a
of
a system subject
to
an
external force with
potential
&(a):[53]
dW
=
A'
e'NRTda,
(I)
where dW
is
the
probability
that the value
of the
parameter
lies
between
a
and
a
+ da,
and A'
a
constant. Einstein
applied eq.
(I)
to
a
harmonic oscillator in
equilibrium
with
a
gas
to derive the
black-body
radiation law in the limit
of
large wavelengths
and
high
tem-
peratures.
An
investigation
of
how small
a particle
must be
in order
to remain
in
suspen-
sion in
a gravitational
field
provides
another
application.
Eq. (I)
does
not, however,
allow the treatment
of
Brownian
motion,
a
time-dependent
process involving
the
interplay
of
fluctuations and
dissipation.
In order to derive
a
fluctua-
tion formula that
generalizes eq.
(1),
Einstein related
a general dissipation
mechanism,
analogous
to
Stokes's
law,
to the condition for
stability
of
the distribution,
eq.
(I),
now
interpreted as giving
the number
of
systems
in
a
certain
state,
rather than the
probability
of
that state. The
potential
I
refers to
a
fictitious
force.[54]
The
resulting
formula for
the
time
dependence
of
the
mean square
fluctuation
(eq.
[II] on
his
p.
378)
enabled Einstein
to treat rotational
as
well
as
translational
motions
of
suspended particles:
[47]
See
Michele
Besso to Einstein,
7-11 Feb-
ruary
1903,
which indicates that there
was
ad-
ditional
correspondence on
this
subject.
See
Sutherland
1897 for
Sutherland's
hypothesis.
[48] See,
in
particular,
Einstein 1902b
(Doc.
3),
§
10.
[49]
At this
time, however,
Einstein
regarded
black-body
radiation
as
the
only physical system
for which
experience suggests
the existence of
observable
energy
fluctuations
(see
Einstein
1904
[Doc.
5],
p.
361).
[50]
Einstein
1906b
(Doc. 32),
p.
371.
[51]
See Einstein 1902b
(Doc. 3),
§
3.
[52]
See Einstein 1904
(Doc. 5),
§
4.
[53]
The choice
of
the observable
parameter,
which
is
treated
by analogy to
the
energy
in
the
fluctuation formula
given
in Einstein 1904
(Doc.
5),
was
the
subject
of
correspondence
between
D.K.C.
MacDonald and Einstein in 1953.
[54]
For
a
discussion
of Einstein's
use
of
such
fictitious forces,
see
the editorial
note,
"Ein-
stein's
Dissertation
on
the
Determination
of
Mo-
lecular Dimensions,"
§
IV,
p.
177.