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 .

2β

47

THE ORNSTEIN-UHLENBECK THEORY OF BROWNIAN MOTION

For t = s this is

σ2

(1 ’ e’2βt ).

2β

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