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chowski, although in agreement with experiment, was clearly a highly
idealized treatment. The theory was far removed from the Newtonian
mechanics of particles. Langevin initiated a train of thought that, in
1930, culminated in a new theory of Brownian motion by L. S. Ornstein
and G. E. Uhlenbeck [22]. For ordinary Brownian motion (e.g., carmine
particles in water) the predictions of the Ornstein-Uhlenbeck theory are
numerically indistinguishable from those of the Einstein-Smoluchowski
theory. However, the Ornstein-Uhlenbeck theory is a truly dynamical
theory and represents great progress in the understanding of Brownian
motion. Also, as we shall see later (Chapter 10), there is a Brownian
motion where the Einstein-Smoluchowski theory breaks down completely
and the Ornstein-Uhlenbeck theory is successful.
The program of reducing Brownian motion to Newtonian particle me-
chanics is still incomplete. The problem, or one formulation of it, is to
deduce each of the following theories from the one below it:

Einstein - Smoluchowski
Ornstein - Uhlenbeck
Maxwell - Boltzmann
Hamilton - Jacobi.

We shall consider the ¬rst of these reductions in detail later (Chapter 10).
Now we shall describe the Ornstein-Uhlenbeck theory for a free particle
and compare it with Einstein™s theory.

45
46 CHAPTER 9

We let x(t) denote the position of a Brownian particle at time t and
assume that the velocity dx/dt = v exists and satis¬es the Langevin
equation
dv(t) = ’βv(t)dt + dB(t). (9.1)
Here B is a Wiener process (with variance parameter to be determined
later) and β is a constant with the dimensions of frequency (inverse time).
Let m be the mass of the particle, so that we can write
d2 x dB
m 2 = ’mβv + m .
dt dt
This is merely formal since B is not di¬erentiable. Thus (using Newton™s
law F = ma) we are considering the force on a free Brownian particle
as made up of two parts, a frictional force F0 = ’mβv with friction
coe¬cient mβ and a ¬‚uctuating force F1 = mdB/dt which is (formally)
a Gaussian stationary process with correlation function of the form a
constant times δ, where the constant will be determined later.
If v(0) = v0 and x(0) = x0 , the solution of the initial value problem
is, by Theorem 8.2,
t
’βt ’βt
eβs dB(s),
v(t) = e v0 + e
0
(9.2)
t
x(t) = x0 + v(s) ds.
0

For a free particle there is no loss of generality in considering only the
case of one-dimensional motion. Let σ 2 be the variance parameter of B
1
(in¬nitesimal generator 2 σ 2 d2 /dv 2 , EdB(t)2 = σ 2 dt). The velocity v(t) is
Gaussian with mean
e’βt v0 ,
by (9.2). To compute the covariance, let t ≥ s. Then
t s
’βt ’βs
βt1
eβs1 dB(s1 )
Ee e dB(t1 )e
0 0
s
= e’β(t+s) e2βr σ 2 dr
0
2βs
’1
’β(t+s) 2 e
=e σ .

47
THE ORNSTEIN-UHLENBECK THEORY OF BROWNIAN MOTION


For t = s this is
σ2
(1 ’ e’2βt ).

Thus, no matter what v0 is, the limiting distribution of v(t) as t ’ ∞ is
Gaussian with mean 0 and variance σ 2 /2β. Now the law of equipartition
of energy in statistical mechanics says that the mean energy of the particle
1
(in equilibrium) per degree of freedom should be 2 kT . Therefore we set

1 σ2 1
m = kT.
2 2β 2
That is, recalling the previous notation D = kT /mβ, we adopt the nota-
tion
βkT
σ2 = 2 = 2β 2 D
m
for the variance parameter of B.
We summarize in the following theorem.

THEOREM 9.1 Let D and β be strictly positive constants and let B be
the Wiener process on ‚ with variance parameter 2β 2 D. The solution of

dv(t) = ’βv(t)dt + dB(t); v(0) = v0

for t > 0 is
t
’βt
e’β(t’s) dB(s).
v(t) = e v0 +
0

The random variables v(t) are Gaussian with mean

m(t) = e’βt v0

and covariance

r(t, s) = βD e’β|t’s| ’ e’β(t+s) .

‚ with in-
The v(t) are the random variables of the Markov process on
¬nitesimal generator
d2
d 2
’βv + β D 2
dv dv
48 CHAPTER 9

2
with domain including Ccom (‚), with initial measure δv0 . The kernel of
the corresponding semigroup operator P t is given by

(v ’ e’βt v0 )2
’1
’2βt
t
p (v0 , dv) = [2πβD(1 ’ e exp ’
)] dv.
2
2βD(1 ’ e’2βt )

The Gaussian measure µ with mean 0 and variance βD is invariant,
P t— µ = µ, and µ is the limiting distribution of v(t) as t ’ ∞.

The process v is called the Ornstein-Uhlenbeck velocity process with
di¬usion coe¬cient D and relaxation time β ’1 , and the corresponding po-
sition process x given by (9.2) is called the Ornstein-Uhlenbeck process.

THEOREM 9.2 Let the v(t) be as in Theorem 9.1, and let
t
x(t) = x0 + v(s) ds.
0

Then the x(t) are Gaussian with mean

1 ’ e’βt
m(t) = x0 +
˜ v0
β

and covariance
D
’2 + 2e’βt + 2e’βs ’ e’β|t’s| ’ e’β(t+s) .
r(t, s) = 2D min(t, s) +
˜
β

Proof. This follows from Theorem 9.1 by integration,
t
m(t) = x0 +
˜ m(s) ds,
0
t s
r(t, s) =
˜ dt1 ds1 r(t1 , s1 ).
0 0

The second integration is tedious but straightforward. QED.

In particular, the variance of x(t) is

D
(’3 + 4e’βt ’ e’2βt ).
2Dt +
β
49
THE ORNSTEIN-UHLENBECK THEORY OF BROWNIAN MOTION


The variance in Einstein™s theory is 2Dt. By elementary calculus, the
absolute value of the di¬erence of the two variances is less than 3Dβ ’1 .
In the typical case of β ’1 = 10’8 sec ., t = 1 sec ., we make a proportional
2
’8
error of less than 3 — 10 by adopting Einstein™s value for the variance.
The following theorem shows that the Einstein theory is a good approxi-
mation to the Ornstein-Uhlenbeck theory for a free particle.

THEOREM 9.3 Let 0 = t0 < t1 < . . . < tn , and let

∆t = min ti ’ ti’1 .
1¤i¤n


Let f (x1 , . . . , xn ) be the probability density function for x(t1 ), . . . , x(tn ),
where x is the Ornstein-Uhlenbeck process with x(0) = x0 , v(0) = v0 ,
di¬usion coe¬cient D and relaxation time β ’1 . Let g(x1 , . . . , xn ) be the
probability density function for w(t1 ), . . . , w(tn ), where w is the Wiener
process with w(0) = x0 and di¬usion coe¬cient D.
Let µ > 0. There exist N1 depending only on µ and n and N2 depending
only on µ such that if

∆t ≥ N1 β ’1 , (9.3)

2
v0
t1 ≥ N2 , (9.4)
2Dβ 2
then

|f (x1 , . . . , xn ) ’ g(x1 , . . . , xn )| dx1 . . . dxn ¤ µ. (9.5)
‚ n




Proof. Assume, as one may without loss of generality, that x0 = 0.
Consider the non-singular linear transformation

(x1 , . . . , xn ) ’ (˜1 , . . . , xn )
x ˜

‚n given by
on
1
xi = [2D(ti ’ ti’1 )]’ 2 (xi ’ xi’1 )
˜ (9.6)

for i = 1, . . . , n. The random variables w(ti ) obtained when this trans-
˜
formation is applied to the w(ti ) are orthonormal since Ew(ti )w(tj ) =
50 CHAPTER 9

2D min(ti , tj ). Thus g , the probability density function of the w(ti ), is
˜ ˜
the unit Gaussian function on ‚ . Let f be the probability density func-
˜
n

tion of the x(ti ), where the x(ti ) are obtained by applying the linear
˜ ˜
transformation (9.6) to the x(ti ). The left hand side of (9.5) is unchanged
˜
when we replace f by f and g by g , since the total variation norm of a
˜
measure is unchanged under a one-to-one measurability-preserving map
such as (9.6).
We use the notation Cov for the covariance of two random variables,
Cov xy = Exy ’ ExEy. By Theorem 9.2 and the remark following it,

Cov x(ti )x(tj ) = Cov w(ti )w(tj ) + µij ,

where |µij | ¤ 3Dβ ’1 . By (9.6),

Cov x(ti )˜(tj ) = δij + µij ,
˜x

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