212
BROWNIAN
MOTION
The
ensuing
derivation
of
the diffusion
equation
is
based
on
the introduction
of
a prob
ability
distribution
for
displacements.
The introduction
of
such
a
distribution
is presum
ably
related to
Einstein's
previous use
of
probability
distributions.[44] Einstein
assumed
the
existence
of
a
time
interval,
short with
respect
to the observation
time,
yet
sufficiently
long enough
that the motions
of
a suspended particle
in
two
successive time intervals
can
be treated
as independent
of
each other. The
displacement
of
the
suspended
particles
can
then be described
by a
probability
distribution that determines the
number of
particles
displaced by a
certain distance
in each
time interval. Einstein
derived
the diffusion
equa
tion
from
an
analysis
of
the
timedependence
of
the
particle
distribution,
calculated
from
the
probability
distribution for
displacements.
This derivation
is
based
on
his crucial
in
sight
into
the
role
of
Brownian motion
as
the
microscopic process
responsible
for diffusion
on
a macroscopic
scale.
Compared
to such
a
derivation, one
based
on
the
analogy
to
the
treatment
of
diffusion in the kinetic
theory
of
gases may
have
appeared more
problematic
to Einstein because
of
the lack
of
a fully developed
kinetic
theory
of
liquids.[45]
The solution
of
the
resulting
diffusion
equation,
combined with his
expression
for
the
diffusion
coefficient,
yields an expression
for the
mean square
displacement,
Xx,
as a
func
tion
of
time
(p. 559), an expression
that Einstein
suggested
could be used
experimentally
to
determine
Avogadro's
number
N:
\x

Vt
V
N
3tt/kP
RT
1
,
(1)
where t is the
time,
R
the
gas constant,
T the
temperature,
k
the
viscosity,
and
P
the
radius
of
the
suspended particles.
Through
his earlier
work,
Einstein
was
familiar with the
theory
of diffusion
in both
gases
and
liquids,
as
well
as
with other
techniques
needed for his
analysis
of Brownian
motion.[46]
In Einstein 1902a
(Doc. 2)
he
suggested
the
replacement
of
semipermeable
walls in
thermodynamic arguments by
external conservative
forces,
a
method he stated to
be
particularly
useful
for
treating arbitrary
mixtures. In 1903 Einstein
discussed
the
Stokes's
formula,
which
is
derived for uniform
motion, to Brownian motion
as
problematic
(see
Smoluchowski 1906,
p. 775;
Perrin
1908b;
and
De
HaasLorentz
1913, pp.
5557).
For
a
dis
cussion
of
this
problem, including references,
see
Fürth
1922,
pp.
5860; fn.
6
to Einstein
1905k
(Doc. 16).
[44]
For
Einstein's
first
use
of
probability
dis
tributions in his
papers on
statistical
physics,
see
Einstein
1902b
(Doc. 3), p.
422.
[45]
For such
a study
of
diffusion in
liquids,
based
on
the
concept
of
mean
free
path,
see
Riecke 1890. In
Einstein
1905j
(Doc.
15),
Ein
stein mentioned the
"insuperable
difficulties
confronting
a
detailed molecularkinetic
theory
of fluids"
("unüberwindlichen
Schwierigkeiten
welche
einer
ins einzelne
gehenden
molekular
kinetischen Theorie der
Flüssigkeiten entgegen
stehen")
(p. 5).
For
evidence
of
Einstein's
skep
ticism with
regard
to
an explanation
of
Brownian motion that follows the methods
of
the kinetic
theory
of
gases,
see
his critical
re
marks
on
Smoluchowski's
work in Einstein to
Carl
Seelig,
15
September
1952,
quoted
in
§
VI.
[46]
For
a
discussion
of Einstein's
earlier inter
est
in
diffusion,
see
the editorial
note,
"Ein
stein's
Dissertation
on
the Determination
of
Mo
lecular
Dimensions,"
pp.
177179.