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where |µij | ¤ 4 · 3Dβ ’1 /2D∆t ¤ 6/N1 if (9.3) holds. Again by Theorem
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

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