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Physics 212: Statistical mechanics II, Fall 2006
Lecture VII
Our ﬁrst picture of Brownian motion will be as a stochastic (i.e., random) process. A heavy
particle in a ﬂuid undergoes random forces as a result of collisions with the light particles of the
ﬂuid. You may be familiar with the idealization of the resulting motion as a random walk, but
since this is a physics course, we will start from Newton’s equation of motion. The assumption of
random forcing leads to a Langevin equation of motion. Our strategy is as follows: we obtain a
constraint on the random forcing from the requirement that it reproduce thermal equilibrium, then
use this to describe the behavior of the system under nonrandom forcing (e.g., an applied electric
ﬁeld to a charged particle).
In writing equations in this lecture, the motion will be assumed to be onedimensional so that
no vector notation is necessary. Including the drag force from Stokes’ law gives:
m
du
dt
=

6
πμau
+
F
(
t
)
≡ 
mγu
+
F
(
t
)
.
(1)
Here we deﬁned
γ
=
6
πμa
m
.
(2)
This equation will simplify depending on our assumptions about
F
. For now we leave it in this
general form.
The ﬁrst goal of this lecture is, given statistical information about
F
(
t
), to carry it through the
Langevin equation (29) to get a statistical picture of
x
(
t
). This sort of random problem appears
frequently in statistical physics: an example of great importance in condensed matter physics is,
given a random potential
V
(
x
), to solve Schrodinger’s equation in that potential and get statistical
information about its eigenstates. There are even many applications beyond physics: stochastic
calculus, the formalization of some ideas in this lecture, is of considerable importance in signal
processing and quantitative ﬁnance, to name just two ﬁelds.
Since the Langevin equation (29) is linear, a natural approach is to take the Fourier transform
of both sides. It will be useful to imagine that we are observing the system for times 0
≤
t
≤
T
, for
some very large time
T
. The reason for doing this rather than just going to
T
=
∞
is that we can
average quantities by 1
/T
and get ﬁnite answers in the limit
T
→ ∞
for quantities like the power
spectrum, even though the underlying functions like
u
(
t
) do not fall oﬀ suﬃciently rapidly at
∞
for the Fourier transform to be well deﬁned. These deﬁnitions will be familiar to any of you with
experience in signal processing. Then the expansion of
u
(
t
) is
u
(
t
) =
X
n
u
n
e
iω
n
t
,
ω
n
=
2
πn
T
.
(3)
and so on for the other quantities in the Langevin equation. The inverse formula is
u
n
=
1
T
Z
T
0
u
(
t
)
e

iω
n
T
.
(4)
The reality of
u
(
t
) then restricts
u
n
*
=
u

n
.
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