188
REVIEW OF BROWNIAN MOTION
§4.
Application of
the
derived
equation
to Brownian motion
Using
equations
(II),
we now
calculate the
mean
displacement
in
a
particular direction
(the
X-direction
of
a
coordinate
system)
experienced
during
time
t
by a
spherical
body
suspended
in
a
liquid.
To
this
end
we
must
substitute the
corresponding
value
for
B
in
the
above equation.
If
a
force is exerted
on a
sphere
of radius
P
that is
suspended
in
a
liquid
with
a
coefficient of friction
k,
the
sphere
will
move
with
velocity1
K/6ikP.
Hence
we
have to
put
B
=
6hcP
'
so
that-in
conformity
with the
paper
cited
above-for the
mean
displacement
of the
suspended
sphere
in the
direction
of
the X-axis
we
obtain the value
tt
_
rr
BT
1
x
&
3S7
*
[24]
Second,
we
consider the
case
when
the
sphere
in
question
is
pivoted
in
the
liquid such
that it
can
freely rotate
(without bearing
friction)
about
one
of its diameters,
and
we
seek
to
determine the
mean
rotation
A2
of the
sphere
produced
by
the
random
thermal
process
during
time t.
If
a
torque
D
acts
upon
a
sphere
of radius
P
that is
pivoted
in
a
liquid
whose
coefficient
of friction is
k,
the
sphere
will
rotate
with the
angular velocity2
D
i) =
Accordingly,
we
have to put
B
=
We
thus
get
1
8¥£F
=
rt
RT
1
,
N
4ir£F"
[23] 1Cf.
G.
Kirchhoff,
Vorles. über
Mechanik
[Lectures
on
Mechanics].
Lecture
26.
2ibid.