Master equation solution of Ornstein-Uhlenbeck processes

Physica 26 485-49 1

Bowen, Julius, I. Meijer, Paul H. E. 1960

MASTER EQUATION SOLUTION OF ORNSTEIN-UHLENBECK PROCESSES *) by JULIUS and PAUL Catholic

University

I. BOWEN

**)

H. E. MEIJER

of America,

Washington,

***) D.C.,

U.S.A.

The continuous master equation is solved in closed form for transition probabilities which are Gaussian and assuming the equilibrium solution to be Gaussian (i.e., the equilibrium fluctuations are Gaussian processes). In this case, the eigenfunctions of the integral equation obtained after the kernel is symmetrized are merely Hermite functions and the eigenvalues are related to one another as sucessive integer powers of a constant 11,, = pn. The constant p is the correlation coefficient for the (stationary) equilibrium process, over the unit time in which the transition probabilities are expressed. The complete (time-dependent) solution for the probability density function is an infinite series of Hermite-type functions, each modified by a term decaying in time. For these Ornstein-Uhlenbeck processes, the relaxation times decrease inversely proportional to the order of the term. The time dependent moments of the distribution of order n can be simply calculated from a knowledge of not more than (IZ + 1)/2 moments of lower order of the initial distribution. Several examples, of different initial distributions, are given.

1. Intro&&on.

a@,t) at

We relate the master equation 1)s) =

to the Smoluchowski

s

P(x’, t) W(x’, x) dx’ -

s

P(x, t) W(x, x’) dx’,

(1)

equation :

P(x, t + 7) = / P(x’, t) V(x’, x ; T) dx’,

(2)

where V(x’, x; t) represents the transition (conditional) probability from x’ to x in the finite time T (independent of t and consequently stationary). Since V is a suitable probability density function, it is normalized for any r: 1 V(x, x’ ; 7) dx’ = where we let the integration *)

This research

was supported

Catholic

University

by the U.S.

26

Air Force

Command. Silver Spring,

of America.

Physica

(3)

domain include x’ = x, to allow for the con-

Research, Air Research and Development **) U.S. Naval Ordnance Laboratory, ***)

1,

485 -

through Maryland.

the Air Force

Office

of Scientific

486

JULIUS

tingency

I. BOWEN

of no transition.

P(x, t + T) -

AND

PAUL

H. E.

7 G

=

-_

Thus t) V(x’, x; T) dx’ -

P(x, t) = /P(x’,

- /P(x, For T sufficiently term

MEIJER

t) V(x, x’; T) dx’.

(4)

short, we replace the left hand side of (4) by the first order

s

P(X’,

t)

v(X’,

X;

T)

dx’ -

s

P(X, t) v(X, x’ ; T) dx’.

(5)

Equation (1) can be obtained from (5) by considering W(x, x’) to be the limit of the quotient V(x’, x; T)/T. Another condition on V is the symmetry relation v(X’, X;

T)

G(x’) = v(x, x’;

T)

G(x),

(6)

where G(x) represents the equilibrium probability density function. This condition is required by the stationarity and reversibility properties of the equilibrium “stochastic process” * ). Condition (6) guarantees that the usual symmetry condition is satisfied : W(x, x’) G(x) = W(x’, x) G(x’).

(7)

Consequently, solutions of (5) for which (6) holds are also solutions of the master equation **) with the usual restriction of detailed balancing automatically imposed. 2. Specification we get

of

Problem.

Assuming P(x, t) = P(x) dt, from (3) and (5)

,uP(x) = j v(x’, x;

T)

P(d)

dx’,

(8)

where ,U = AT + 1. (Note : we anticipate of course that 1 will be a constant, independent of 7, at least in the limit of short T.) We can in the usual way 3, obtain a symmetric kernel for this integral equation: Define zI(X’, X; be the variable

joint

is in the neighbourhood

T)

probability

=

v(X’,

density

X;

7)/G(X),

function

(9)

(at

equilibrium)

that

of x and ti, and of x’ at ts. Then, using the properties

the

random

of stationarity

and reversibility:

**) Equation (1) also implies that V(x’, x; 7) is linear in Q-in the limit of r approaching zero, so that W(x’x) independent of 7 exists. (For this reason, the integrations of (1) are usually taken to mean principal value integrals.) This linearity is not assumed in the solution of (5) but it is instead assumed

that

P(x, t + T) -

apb, t)

P(x, t) = 7 - __

at

MASTER

EQUATION

SOLUTION

OF ORNSTEIN-UHLENBECK

where by (6) v(x’,x; T) is symmetric

PROCESSES

487

in x, x’. Then define

Q(x)= P(x)(@))-~,

(10)

and K(x’, x; T) = v(x’, x; ~)(G(x) G(x’))*.

(11)

pQ(4 =/W',

(12)

Thus (8) becomes

x;7)Q(x') dx',

and K(x’, x; T) is symmetric in x’, x. For the equilibrium G(x), we will consider here G(x) = (al/%)--r

exp - +(x/0)2,

(13)

in common with many other investigators 4)s). In this form G(x) is already normalized. We wish now to select a conditional probability density function (p.d.f.) V(x, x’;T) which satisfies (3) as well as (6). Clearly one such case is C exp - a[x’ - b(~)x]s, since this is Gaussian in x’, with mean bx, and one can normalize i.e., by integrating over x’ and adjusting the constant C. The constants a and 6 can be selected to satisfy detailed balancing. Using (13) to specify G(x) : 1 1 v(x, Cd; T) = (14) __ (x’ - px)2, exp 2a2( 1 - p2) aA&1 - $9” where p = p(T). This is the conditional p.d.f. associated with a Gaussian random process. It will be recognized that this case has long been studied in the theory of Brownian motion and the process (equilibrium process) described is called the Ornstein-Uhlenbeck process 6)7)s). The quantity p has a well known meaning in statistics and in physics: it is the correlation function, specifying the expected value of xx’: pas = E(xx’) = Jxx’ fs dx dx’.

(15)

We will return to this discussion after solving (12) for this case. 3. Solution of the Integral Equation. With the above-mentioned of G(x), V(x, x’; T), we find from (1 1): 1

K(x’, x) = adsc(

-- exp 1-p2)”

We note upon direct substitution 1

pW-2pxx’+p2x2

x2

202( 1 - p2)

x12 1

choices

+- 4a2 +GqJ

(16)

(x’ - pr)2] .

(17)

that

x12

-

1

-

[ 2a2( 1 -

p2)

488

JULIUS

A useful expansion exp -

1 ___-__ [ 2a2( 1 =2/l

I. BOWEN

AND

PAUL

H. E. MEIJER

formula for the above exponential

p2)

px)“]=

(x’ -

c1

.___-p2exp-

5o H”(~‘/dwHY(4l/W

xl2 2oz

where Hn(4

is a) :

Y!

1) n exp [ts](d/dt)” exp -

= (-

P

lJ

(->2

’

(18)

[t2],

the Hermite polynomial 10). Now defining the Hermite function &(t) = = Hn(t) exp - [P/2], these &(t) form an orthogonal sequence (with unit weight factor) : J-“oo h&) One can verify,

&(t)

using equations

dt = S,,

2/n2”n!.

(16) through

(19)

( 19), that

s

(20)

Thus we have the solution, en(X)

=

with the eigenfunctions

hn

(5) ;

pn

=

An7

i.e. the eigenfunctions are simply the Hermite The solution of the master equation is p(x,

t) =

5

+

and eigenvalues:

1=

pnj

functions.

e-.+/2o2 &"-l)t/:.

A&,

n=O

Note that lo = 0, 1, < O(n f from initial conditions : An=

1

a4G2”n

0). The constants

!

(21)

A, are of course evaluated

s

(22)

Since Ha is a polynomial of order n and its argument has only even or only odd powers, A n is determined by either n/2 or (n + 1)/2 moments of the initial distribution, in accordance with whether n is even or odd. It is readily seen that P(x, t) is automatically normalized for all times because, upon integration of (21) one obtains (using Ho(t) = 1):

Z:n

An

exp[(pn-

l)t/Tl~~m (-$) Hn (&) Ho

exp-

$ dx=Aoa@Z

and by (22), A0 = 1/oz/2z as long as P(x, 0) is normalized. Thus /P(x, t) dx = 1, for all times. The time-dependent moments ,un(t) = /xnP(x, t) dx can easily

be ex-

MASTER

EQUATION

SOLUTION

OF ORNSTEIN-UHLENBECK

PROCESSES

489

pressed in terms of the solution (21). This can be done by expressing xn as a linear combination of Hermite polynomials up to and including that of order n, and using the orthogonality (19) to perform the integration. This leads to a finite number of terms with constants An, which are in turn simply obtained from the moments of the initial distribution. Consequently, the time dependent moments are completely determined by a finite number of moments of the initial distribution, and a detailed knowledge of the initial distribution is not required. For example, it can easily be shown that pi(t) = ~(0) p2(t)

=

a2 +

WI

exp[(p -

[p2(0)

-

;

(23)

a21exp[(p2 - l)+l.

(24

4. Relaxation Times for the Omstein-Uhlenbeck Process. On the basis of the Langevin equation 6), frequently taken as the starting point for the discussion of such processes, one obtains: p(7) = exp -

Y 171

(25)

where y is a positive constant. This starting point is equivalent to the assumption that (13) is the applicable p.d.f. Then, according to Doob s) one gets again (25). Doob’s theorem requires that the process in question be stationary (as at equilibrium) and Markoffian, and that x and x’ have a bivariate Gaussian distribution. From (25) one gets 1, = (epnyT -

1)/r --f 740

(26)

ny,

i.e., relaxation times associated with higher order terms in P(x, t) vary inversely with n, the order. In this limit the relaxation-times are independent of T, as mentioned in the beginning of section 2. A similar result was obtained by Van K amp en’s) from the solution of a Fokker-Planck type of differential equation, derived from (1) after some assumptions of the probability density functions. 5. Examples. any distribution the form

about the sharpness

1. It can be demonstrated from the solution obtained that function which is initially Gaussian remains, so and has

P(x, t) = [a(t) 1/Zi-1

exp -

where x(t) = x 02(t) = p2(t) -

PI(t) = x -

8

x(t) 2 __ ( a(t) > ’

(27)

x0 e+;

[q(t)12 = ~2 + [a$ -

a2] ewzyt,

x0 and (TOcorresponding to the initial mean and variance. The initial Gaussian distribution remains Gaussian for all times because of the form of the

490

JULIUS

I. BOWEN

AND

PAUL

H. E. MEIJER

transition p.d.f. (14), as can be seen from the Smoluchowski equation. The Gaussian in turn is completely determined by only two moments. This result (27) allows us to consider the evolution of a distribution which originally is kept at mean x0 with an external influence, one with an initial temperature different from that of final equilibrium, or both simultaneously. 2. Consider a system consisting of one-dimensional gas with a ‘Maxwell distribution of velocity x, at equilibrium (in contact with a bath) at a temperature corresponding to cr2, At a certain instant of time a shutter is opened which allows gas particles at equilibrium with temperature aasto enter.

Fig. 1. Approach

of

the

Rayleigh

distribution

to

the

Boltzmann

distribution

for

so/a = 4.

The shutter is closed after a short time, i.e., before the incoming gas can undergo any appreciable transitions. The incoming gas contains only particles with positive velocity, x > 0, and has the (Rayleigh) p.d.f. P(x, 0) = (x/IJ$) exp -

Q(x/aa)s,

x>

0.

The resulting formulae for the coefficients have been evaluated for the case a/a0 = 2, and the resulting p.d.f., appropriately non-dimensional and for various times are plotted in figure 1.

MASTER

EQUATION

6. Discussio~z.

Far

SOLUTION

OF ORNSTEIN-UHLENBECK

the Gaussian

transition

PROCESSES

probability,

491

which satisfies

detailed balancing, one can solve the linear master equation exactly and in closed form. It is well known that this transition probability has played a central role in historical development of Brownian motion theory. The Langevin equation and the usual Fokker-Planck equation lead to such a p.d.f. and this holds as well when one treats the Langevin equation rigorous1Y 9. There are two other interesting points of view we should like to mention: Keilson and Storer 1) show how certain (physical) assumptions, slightly different than the Langevin equation restrict one to the consideration of Gaussian processes only. Van K am pen, on the other hand, has derived a Fokker-Planck type of equation from the master equation. In a specific example 9, the coefficients appearing in the equation, which are certain moments of the transition p.d.f., are evaluated by comparing the physical system under study with the known phenomonological behavior. The solution of the resulting equation is similar to the solution obtained herein and the Gaussian process is required by the solution. Similarly one would expect that the undetermined constant y, which appears in this paper as the inverse of the relaxation time associated with the first moment, could be evaluated from the phenomonological behaviour of an appropriate physical system. Received 3-12-59.

REFERENCES

1) 2)

Keilson,

J. and Storer,

J. E., Quart. Journ. appl. Math. 10 (1952) 243.

Van Kampen, N. G., Physica 23 (1957) 707. 3) Meijer, P. H. E. and Bowen, J. I., preceding paper. L. aud Machlup, S., Phys. Rev. 91 (1953) 1505. 4) Onsager, N. G., Phys. Rev. 110 (1958) 319. 5) Van Kampen, G. E., and Ornstein, L. S., Phys. Rev. 38 (1930) 823. 6) Uhlenbeck, G. E., Rev. mod. Phys. 17 (1945) 323. 7) Wang, M. C. and Uhlenbeck, 8) I>oob, J. L., Ann. Math. 43 (1942) 351. 9) Cramer, H., Mathematical Methods of Statistics, chapt. 12, sect. 6. 10) Schiff,

L. I., Quantum

Mechanics

(Princeton University Press, Princeton, 1946).

(McGraw-Hill Co., N.Y.,

Second Edition,

1955).