9.2, the mean of x(t1 ) is, in absolute value, smaller than

˜

’1

1

|v0 |/β[2Dt1 ] ¤ N2 2

2

if (9.4) holds. The mean of x(ti ) for i > 1 is, in absolute value, smaller

˜

than

1

(e’βti’1 ’ e’βti )|v0 |/β[2D(ti ’ ti’1 )] 2 .

Since the ¬rst factor is smaller than 1, the square of this is smaller than

e’βti’1 ’ e’βti v0 N1 e’N1

2

βt1 ’βt1

¤ ¤

e

2Dβ 2

ti ’ ti’1 N2 N2

if (9.3) and (9.4) hold with N1 ≥ 1. Therefore, if we choose N1 and N2

˜

large enough, the mean and covariance of f are arbitrarily close to 0 and

δij , respectively, which concludes the proof. QED.

Chandrasekhar omits the condition (9.4) in his discussion [21, equa-

tions (171) through (174)], but his reasoning is circular. Clearly, if

v0 is enormous then t1 must be suitable large before the Wiener pro-

cess is a good approximation. The condition (9.3) is usually written

β ’1 (∆t much larger than β ’1 ). If v0 is a typical velocity”i.e., if

∆t

1 1

|v0 | is not much larger than the standard deviation (kT /m) 2 = (Dβ) 2

51

THE ORNSTEIN-UHLENBECK THEORY OF BROWNIAN MOTION

of the Maxwellian velocity distribution”then the condition (9.4), t1

β ’1 .

v0 /2Dβ 2 , is no additional restriction if ∆t

2

There is another, and quite weak, formulation of the fact that the

Wiener process is a good approximation to the Ornstein-Uhlenbeck pro-

cess for a free particle in the limit of very large β (very short relaxation

time) but D of reasonable size.

DEFINITION. Let x± , x be real stochastic processes indexed by the same

index set T but not necessarily de¬ned on a common probability space.

We say that x± converges to x in distribution in case for each t1 , . . . , tn in

T , the distribution of x± (t1 ), . . . , x± (tn ) converges (in the weak-— topology

of measures on ‚n , as ± ranges over a directed set) to the distribution of

x(t1 ), . . . , x(tn ).

It is easy to see that if we represent all of the processes in the usual

way [25] on „¦ = ‚ , this is the same as saying that Pr± converges to Pr

™I

in the weak-— topology of regular Borel measures on „¦, where Pr± is the

regular Borel measure associated with x± and Pr.

The following two theorems are trivial.

THEOREM 9.4 Let x± , x be Gaussian stochastic processes with means

m± , m and covariances r± , r. Then x± converges to x in distribution if

and only if r± ’ r and m± ’ m pointwise (on T and T — T respectively,

where T is the common index set of the processes).

THEOREM 9.5 Let β and σ 2 vary in such a way that β ’ ∞ and

D = σ 2 /2β 2 remains constant. Then for all v0 the Ornstein-Uhlenbeck

process with initial conditions x(0) = x0 , v(0) = v0 , di¬usion coe¬cient

D, and relaxation time β ’1 converges in distribution to the Wiener pro-

cess starting at x0 with di¬usion coe¬cient D.

References

The best account of the Ornstein-Uhlenbeck theory and related mat-

ters is:

[21]. S. Chandrasekhar, Stochastic problems in physics and astronomy,

Reviews of Modern Physics 15 (1943), 1“89.

See also:

52 CHAPTER 9

[22]. G. E. Uhlenbeck and L. S. Ornstein, On the theory of Brownian

motion, Physical Review 36 (1930), 823“841.

[23]. Ming Chen Wang and G. E. Uhlenbeck, On the theory of Brownian

motion II, Reviews of Modern Physics 17 (1945), 323“342.

The ¬rst mathematically rigorous treatment, and additionally the

source of great conceptual and computational simpli¬cations, was:

[24]. J. L. Doob, The Brownian movement and stochastic equations, An-

nals of Mathematics 43 (1942), 351“369.

All four of these articles are reprinted in the Dover paperback “Se-

lected Papers on Noise and Stochastic Processes”, edited by Nelson Wax.

[24]. E. Nelson, Regular probability measures on function space, Annals

of Mathematics 69 (1959), 630“643.

Chapter 10

Brownian motion in a force

¬eld

We continue the discussion of the Ornstein-Uhlenbeck theory. Suppose

we have a Brownian particle in an external ¬eld of force given by K(x, t) in

units of force per unit mass (acceleration). Then the Langevin equations

of the Ornstein-Uhlenbeck theory become

dx(t) = v(t)dt

(10.1)

dv(t) = K x(t), t dt ’ βv(t)dt + dB(t),

where B is a Wiener process with variance parameter 2β 2 D. This is of

the form considered in Theorem 8.1:

v(t)

x(t) 0

d = dt + d .

K x(t), t ’ βv(t)

v(t) B(t)

Notice that we can no longer consider the velocity process, or a component

of it, by itself.

For a free particle (K = 0) we have seen that the Wiener process,

which is a Markov process on coordinate space (x-space) is a good approx-

imation, except for very small time intervals, to the Ornstein-Uhlenbeck

process, which is a Markov process on phase space (x, v-space). Similarly,

when an external force is present, there is a Markov process on coordinate

space, discovered by Smoluchowski, which under certain circumstances is

a good approximation to the position x(t) of the Ornstein-Uhlenbeck pro-

cess.

Suppose, to begin with, that K is a constant. The force on a particle

of mass m is Km and the friction coe¬cient is mβ, so the particle should

53

54 CHAPTER 10

acquire the limiting velocity Km/mβ = K/β. That is, for times large

compared to the relaxation time β ’1 the velocity should be approximately

K/β. If we include the random ¬‚uctuations due to Brownian motion, this

suggests the equation

K

dx(t) = dt + dw(t)

β

where w is the Wiener process with di¬usion coe¬cient D = kT /mβ.

β ’1 ,

If there were no di¬usion we would have, approximately for t

dx(t) = (K/β)dt, and if there were no force we would have dx(t) = dw(t) .

If now K depends on x and t, but varies so slowly that it is approximately

constant along trajectories for times of the order β ’1 , we write

K x(t), t

dx(t) = dt + dw(t).

β

This is the basic equation of the Smoluchowski theory; cf. Chandrasek-

har™s discussion [21].

We shall begin by discussing the simplest case, when K is linear and

independent of t. Consider the one-dimensional harmonic oscillator with

circular frequency ω. The Langevin equation in the Ornstein-Uhlenbeck

theory is then

dx(t) = v(t)dt

dv(t) = ’ω 2 x(t)dt ’ βv(t)dt + dB(t)

or

x x 0

0 1

d = dt + d ,

2

’ω ’β

v v B

where, as before, B is a Wiener process with variance parameter σ 2 =

2βkT /m = 2β 2 D.

The characteristic equation of the matrix

0 1

A= 2

’ω ’β

is µ2 + βµ + ω 2 = 0, with the eigenvalues

1 12 1 12

β ’ ω2, β ’ ω2.

µ1 = ’ β + µ2 = ’ β ’

2 4 2 4

BROWNIAN MOTION IN A FORCE FIELD 55

As in the elementary theory of the harmonic oscillator without Brownian

motion, we distinguish three cases:

overdamped β > 2ω,

critically damped β = 2ω,

underdamped β < 2ω.

Except in the critically damped case, the matrix exp(tA) is

µ2 eµ1 t ’ µ1 eµ2 t ’eµ1 t + eµ2 t

1

tA

e = .

µ2 µ1 eµ1 t ’ µ2 µ1 eµ2 t ’µ1 eµ1 t + µ2 eµ2 t

µ2 ’ µ1

(We derive this as follows. Each matrix entry must be a linear com-

bination of exp(µ1 t) and exp(µ2 t). The coe¬cients are determined by

the requirements that exp(tA) and d exp(tA)/dt are 1 and A respectively

when t = 0.)

We let x(0) = x0 , v(0) = v0 . Then the mean of the process is

x0

etA .

v0

0

The covariance matrix of the Wiener process is

B

0 0

2c = .

0 2β 2 D

The covariance matrix of the x, v process can be determined from Theo-

rem 8.2, but the formulas are complicated and not very illuminating. The

covariance for equal times are listed by Chandrasekhar [21, original page

30].

The Smoluchowski approximation is

ω2

dx(t) = ’ x(t)dt + dw(t),

β

where w is a Wiener process with di¬usion coe¬cient D. This has the

same form as the Ornstein-Uhlenbeck velocity process for a free parti-

cle. According to the intuitive argument leading to the Smoluchowski

equation, it should be a good approximation for time intervals large com-

β ’1 ) when the force is slowly varying

pared to the relaxation time (∆t

(β 2ω; i.e., the highly overdamped case).

56 CHAPTER 10

The Brownian motion of a harmonically bound particle has been in-

vestigated experimentally by Gerlach and Lehrer and by Kappler [26].

The particle is a very small mirror suspended in a gas by a thin quartz

¬ber. The mirror can rotate but the torsion of the ¬ber supplies a linear

restoring force. Bombardment of the mirror by the molecules of the gas

causes a Brownian motion of the mirror. The Brownian motion is one-

dimensional, being described by the angle that the mirror makes with its

equilibrium position. (This angle, which is very small, can be measured

accurately by shining a light on the mirror and measuring the position

of the re¬‚ected spot a large distance away.) At atmospheric pressure the

motion is highly overdamped, but at su¬ciently low pressures the under-

damped case can be observed, too. The Ornstein-Uhlenbeck theory gives

for the invariant measure (limiting distribution as t ’ ∞) Ex2 = kT /mω 2

and Ev 2 = kT /m. That is, the expected value of the kinetic energy 1 mv 2

2

1

in equilibrium is 2 kT , in accordance with the equipartition law of statis-

tical mechanics. These values are independent of β, and the constancy

of Ex2 as the pressure varies was observed experimentally. However, the

appearance of the trajectories varies tremendously.

Consider Fig. 5a on p. 243 of Kappler [26], which is the same as Fig. 5b

on p. 169 of Barnes and Silverman [11, §3]. This is a record of the motion

in the highly overdamped case. Locally the graph looks very much like

the Wiener process, extremely rough. However, the graph never rises

very far above or sinks very far below a median position, and there is

a general tendency to return to the median position. If we reverse the

direction if time, the graph looks very much the same. This process is a

Markov process”there is no memory of previous positions. A graph of

the velocity in the Ornstein-Uhlenbeck process for a free particle would

look the same.

Now consider Fig. 6a on p. 244 of Kappler (Fig. 5c on p. 169 of Barnes

and Silverman). This is a record of the motion in the underdamped

case. The curve looks smooth and more or less sinusoidal. This is clearly

not the graph of a Markov process, as there is an evident distinction