Stochastic processes with colored Gaussian noise: The small noise limit revisited

Volume 139, number 1,2

PHYSICS LETTERS A

24 July 1989

STOCHASTIC PROCESSES WITH COLORED GAUSSIAN NOISE: THE SMALL NOISE LIMIT REVISITED ~ J. LUCZKA Department of Theoretical Physics, Silesian University, 40-OO7Katowice, Poland Received 12 December 1988; revised manuscript received 17 May 1989; accepted for publication 22 May 1989 Communicated by A.R. Bishop

A non-Markovian (integro-differential) master equation for the probability distribution of stochastic processes driven by colored Gaussian noise is re-considered in the limit of small intensity of the noise. In the literature one can find two differential (“Markovian”) approximative equations to the integro-differential equation. These equations turn out to be incorrect due to some formal but inadequate and uncontrolled approximations. In the paper we construct a novel Markovian approximation. Two widely used models are applied to verify the correctness ofthe three approximations. It is shown that only the novel approximalion is correct.

1. Introduction A great deal of interest has been devoted to the study of dynamical nonlinear systems driven by linear noises and modelled by the stochastic equation p(x, 0) =p(x)

,

xe (x1, x2)

(1)

wheref(x) and g(x) are given functions, p(x, 0) is an initial probability distribution of the process x~, ~(t) is colored Gaussian noise [1], d~(t) =

—

a~(I) dt+ a\/~d W(t),

(~(0)>=0,

<~2(0)> =a,

(2)

(W(t) is the standard process),of~ the is a noise weak 2 Wiener is the intensity coupling constant (e From eq. (2) it follows that the probability distribution po(~,t) of the noise ~(t) does not depend on time and has the Gaussian form 2 1 2 p 0(~,t)=p0(~)=(l/2ira)’~exp(—~/2a), (3) and ~ Work supported in part by the Programme CPBP 01.03.

<~(t)~(s)> =aexp(—alt-—sI)

(4)

The parameter r= 1/a is the correlation time of the noise. Because in a general case there does not exist an analytical theory for (1) and the simple probabilistic characteristics of x~are not known exactly, many approximative theories have been elaborated (it is impossible to quote all papers, see, e.g., refs. [2—5]and references therein). A common feature of almost all approximate theories is that the evolution equation for a single-event probability distribution p (x, I) of the process x1 is approximated by Fokker— Planck-type equations with modified diffusion functions (cf. refs. [2—4] and references therein). But any approximative damental condition: theory should satisfy one fun(a) p(x, t) should be definite and non-negative definite on the whole phase-space (x 1, x2) of the systern (I). In particular: t (x) (if it exists) (b) the stationary distribution pS should satisfy the property (a). In such a case we will say that the theory is mathematically acceptable. For approximate Fokker— Planck-type equations we have a simple criterion: (c) the modified diffusion function should be def-

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mite and non-negative definite on the whole phasespace (x1, x2) of the system (1). The non-Markovian process x, can be approximated by the Markovian (in particular, diffusion) process only in some limiting cases [6—8].Ifr=l/ a = 0 then the noise ~(t) is white and x~is a diffusion process. If z is sufficiently small then x~differs little from the diffusion process defined by (1) when t = ü and it seems that in such a case x~can be approximated by a Markovian (diffusion) process [6—8].If an approximation procedure is correctly constructed then an approximative probability distribution of the process x1 should satisfy the conditions (a) and (b) for some restricted domain of the noise parameters ~ and t (e.g., small and small x). It cannot be required and expected that any mathematically acceptable approximation should be correct for all values of ~ and r. As an example, let us consider a bistable system (1) with 3 (a,b>0), f(x)=ax—bx g(x)=l, XE(—~,~). (5) The conventional small-noise-correlation-time theory [9] leads to the diffusion function

24

version of this latter. The three approximations are tested with two examples discussed in the literature and for them only our novel approximation is mathematically acceptable.

2. A master equation By the use of several different techniques [8,12— 161, one can derive an exact evolution equation for the probability distribution p(x, t) of the process x, in (1). It has the well-known form [4,8,12—151 a (x 1) ~ = LaP (x, 1) + ~2 ds HE ( t s )p (x, s). (8)

j

~‘

The integral kernel

J

=

(6)

(10)

Lo=La+Lr,,

La~~f(X),

Lrt~+ct2~,

D

[l]

30

(9)

L=L

L, = —~-~g(x).

satisfied and the theory is acceptable. Similarly, the unified theory [2] requires that ar< 1. In the present paper we recall an exact integro-differential (in time) equation for p(x, t) of the process x( in (1) that has been derived by many procedures [8,12—15].The principal object is to obtain approximative differential equations in some limiting cases. Two such equations have been reported in the literature [8,1 3—15]. These equations are incorrect because of formal but wrong approximations. We construct a novel approximate differential (diffusion) equation to the integro-differential equation. The procedure used is based on a well-established mathematical method, namely, on an integral

has the form [4]

In eqs. (8) and (9)

D1(x)= and is not ac,ceptable because the condition (c) is not fulfilled. The “best Fokker—Planck-type approximation” [10] is also unacceptable. Fox’s small-correlation-time theory [11] gives the diffusion function 2(1—at+3btx2)’, (7) 2(x)= and requires that at< 1. Then the condition (c) is

HE(t)

—

d~LieQL!Lipo(~).

0+fL1,

2(1+ax—3btx2),

July 1989

(II)

ax The projection operator P is defined by the relation

PA(x,~.t)=p 0(~)

d~A(x,~.t),

(12)

for an arbitrary function A (x, t) and Q = 1 P. Using the relation 2It=e~0Texp due~’~°~’QL e eQ~~0~ 1 (13) / ~,

—

(~J

where T is the time-ordering operator, it is seen that the kernel HE(t) admits a power series expansion in ,

HE(t) =H0(t)+(H1(t)+

2H 2(t)

+...

.

(14)

One argues [4,8,13—19]that if is small enough one

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might neglect all higher-order terms keeping only l(0(t). Then eq. (8) is approximated by [8,14—16] ôp(x, t)

Lap(X,

at

t)

+(2

5

dsH0(t—s)p(x, s),

0

(15)

in which Ho(t)=ae~t~g(x) &t~g(x),

(16)

tativity

t 0(~)

(17)

=ae~

has been utilized, A further simplification is obtained if the memory effects which are present in (15) can be neglected. One expects such a situation when the time scale Ta on which p(x, t) varies is much longer than the decay time of the kernel H 0 (t), that is, than the decay time r= I/a of the correlation function (4) of the noise (Ta>> r) scale) [7]. For and large t (that is 2t-time we small then obtain the Born—Maron the kovian approximation to the non-Markovian master equation (8) as ôp(x,

t)

convolution

in

(15)

and

the

p(x, t—s) ~p(x, t) (20) are used in refs. [15,16] to obtain (18) with (19). Let us notice that eq. (18) with L = ~ is a differential equation of infinite order in x. 4. The second approximation If ~= 0 then from (8) or (15) it follows that

t~ de~

of

approximation

and the relation

5

24 July 1989

Lp(x t)+21p(x,

t) .

p(x, =eLa~_~p(x, t) . for small One s) argues [8,14] that

~,

(21) p(x, s) in (15)

can be approximated by eq. (21) and on the c2t-time scale ~2=

JduHo(u)e_Lau

.

(22)

0

This approximation is diffusion-type and eq. (18) with ~=~2 is a Fokker—Planck equation which coincides withThis theequation “best Fokker—Planck-type tion” [10]. has been derived by equaother methods [5,9,17—19] and has been widely used in the literature.

(18)

at

5. The third approximation

By similar physical arguments, in the literature one can find two different approximations and different generators L in (18). Now, we will recall these approximations.

Now, we will construct a novel approximative differential equation to the integro-differential equation (15). Our method is based rather on some mathematical (natural) arguments than physical (intuitional) ones (cf. (20) and (21)). Indeed, an

3. The first approximation

integral version of eq. (15) leads to a new Markovian approximation. Going over to the interaction

In refs. [15,16] it is argued that the s-integration in (15) effectively extends only over the interval (t—s, t). Assuming p(x, t) varies slowly over this interval, (15) may be further reduced to (18) with L=L~,where

picture ~(x, t)=e

alp(x, t)

(23)

,

from eq. (15) we have ô~(x,I)

5

2e~” duH

at

0(t—u) e~p(x,u).

(24)

0

ii

=

~

(19)

We write eq. (24) as an integral equation

0

Practically, it means that the property of commu31

Volume 139. number 1,2

~(x,

t)

=p

5

+2

PHYSICS LETTERS A

(x)

24 July 1989

6. Verification of the approximations

dse’~

1)

5

duH0(s—u) eLaup~(x,u)

We should decide one of the three approximations, l1, 12 or 13,which is mathematically acceptable. Let us consider two widely used models (1) with [5,20]

0

f(x)=ax—bx~

=P(x)+2fdue_Lau

,

g(x)=x,

xE[0,cc),

(32)

0

where a, b and y are positive parameters, and the

i_u

x

5

(25)

dze_LHo(z)e~~p~(x,u),

model (5). From (19) and (16) one gets

0

where we have changed the integration order and substituted s=z+u. Let us rescale the time I’ =e~t and denote ~(x,t) =J3(x,

t

/~)—p~(x,t’).

Then from (25) one gets

+

5

due

a~21~(t

(26)

is fixed)

(t’

—u, ) e~2pE(x,u),

(27)

=

~

(l+r~f(x))~

~g(x),

(33)

and a stationary solution of eq. (18) with 1=1 has the form t(x) 2g2(x)—rf2(x) N p~ 1g(x) ~-v f(x’) (34) where N 1 is a normalization constant.

0

where

5

1/12

13(t,)=

the~H0(s).

(28)

0

2 can

One can check that p~~t(x) is not defined on the whole phase-space [0, cc) for the model (32) and on (—cc, cc) for the model (5). So, the first approximation is not correct. To find 12 we need (cf. eqs. (22) and (16)) G(t) =e’~’~-g(x) ~

(35)

Forextended sufficiently small ~the upsense to tR be to infinity andintegration in a formal Tofindl 13= urn 13 (t,

~)=

J

ds e

L~sH0

(s).

(29)

0

ax

So, eq. (27) is approximated by p1(x, t’

+

5

) z=p(x) 2L duel

2p 3e

~/E

1(xu).

(30)

o

The differential form of this equation is 21 2~,(x, t’) a~(x, at’ t’ =e’~’~” e31~~E .

ax

where the function F(x, t) is a solution to the partial differential equation [51 ÔF(x, I) = df(x) F(x, t) —f(x) aF(x, 1) at dx ax’ F(x, 0) =g(x) (37) Firstly, let us considerthe model (32). For this model .

(31)

Using eq. (26), from eq. (3l)we obtain eq. (18) with our new Markovian which1=13. is alsoIt ais diffusion-type one. approximation 32

3weneed (cf.eqs. (29) and (16)) G(—t). From ref. [10] it follows that G(t) = F(x, t), G( —I) = ~F(x, —1), (36)

(38) a a Eq. (38) presents a deterministic time-evolution of

F(x,t)_X_~x}+~xie_2i~.

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the operator (ô/8x)g(x) and therefore ya in (38) can be identified with the decay time Ta = 1 /ya. The condition Ta>> r is then equivalent to the condition

24 July 1989

Using the results of ref. [101 gives the generator 12 in the form 2

(46)

2L2=~~_~132(x),

yaT<< 1 . (39) From eqs. (36), (38), (22) and (16) we have

ax where (cf. eq. (4.8) in ref. [10])

(40)

~2(x)=_~__F(_~,1;~(l/aT+l),fl2x2), 1—aT

ybT

ax ax and

13

l+yar

(47)

in (29) takes the form

ax

1

(F is a hypergeometric function) and at< 1. From

ybT x’i+’)_~_x. yar ôx

(41)

—

(36), (45) and (29) it follows that the third generator 13 takes the form 2L ~ -f-. (48) ~ ax 3(x) ax’ —

Eq. (18)inwithin these two approximations can be rewritten the Fokker—Planck—Kolmogorov—Stratonovich form

ap(x,t)

where

a

2

at +

=—~~(x)p,(x,t) ~g(x) ~g(x)D 1(x)p,(x, t),

ax

i=2, 3,

ax

42

where ~(x)=f(x),

D

ybT

2(x)=~2(l_

2x2).

1+yaT~)’ (43)

and ]~(x)=f(x)+g2(x)D~(x),

Eq. (18) with 12 in (46) and 13 in (48) can also be rewritten in the form (40) with g(x)= 1. The function (49) satisfies the condition (c) if 2at< 1. E.g., forr=l/4a(cf.formula(125),p.462,inref. [21]) 2x2)512 (50) 4~2 l—(fl 2x2 D3(x)=~ l—fl The function (47) does not satisfy the condition (c). E.g., if T= l/3a then from (47) it follows that (cf. formula (125), p.462, in ref. [21])

D

2x2 [l—(I—fl2x2)512]. 3(X)=2(l+ l_yaTx),

(44)

andf(x), g(x) are given by eq. (32) and the prime denotes a derivative with respect to x. The function 3 .1 2(x) does not satisfy the condition (c). In consequence, the “best Fokker—Planck-type approximation” should be ruled out for the model (32). In the third approximation, D3(x) is positive definite for all xe [0, cc) under the condition (39). So, the third approximation is mathematically acceptable for (32). Now, let us consider the model (5). In this case, a solution of eq. (37) has the form / 2~~t\ It’A2~23/2 F~ t~—e’°1— ~1 _e_ fl2=b/a>0 (45) —

(49)

D3(x)= I —2aT ~ F(—~, 1; l/2aT, l—fl

‘

(51)

D2(x)= 5fl In fact, eq. (47) is correct only if /ixI <1. If jflx~>1 then .15 2(x) does not exist (more precisely, 5 1 2(x)=cc) ~. Thus, the second approximation is incorrect for the model (5) as well, while our approximation is mathematically acceptable. Let us stress that our approximation is based on the small noise assumption. On the other hand, the Fox theory [11] or the unified theory [2] assume the small correlation time of the noise. Therefore generally they are not comparable with ours. E.g., if we compare corresponding stationary probability III

Many formulas in ref. [10] are incorrect or not precise. E.g., eqs. (3.8), (3.12) and (A6) therein are incorrect. A correct form of these equations is presented in ref. [19]. 33

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distributions for the model (32) then they are very close for small ( and small t (the author performed such a numerical analysis). Finally, let us mention that the problem considered is somewhat similar to that for quantum open systems (see, e.g., refs. [22,23]).

Acknowledgement The author would like to thank the referees for remarks.

References [1]C.W. Gardiner, Handbook of stochastic methods (Springer, Berlin, 1983). [2]P. Jung and P. Hanggi, Phys. Rev. A 35(1987) 4464. [3]E. Peacock-Lopez, B.J. West and K. Lindenberg, Phys. Rev. A 37 (1988) 3530. [4]S. Faetti, L. Fronzoni, P. Grigolini andR. Mannella, J. Stat. Phys. 52(1988)951.

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24 July 1989

[5] J. Luczka, Physica A 153 (1988) 619. [61R.L. Stratonovich, Conditional Markov processes and their application to optimal control (Elsevier, Amsterdam, 1968). [7] VI. Tikhonov and M.A. Mironov, Markovian processes (Sovietskoe Radio, Moscow, 1977) [in Russian 1. [8]M. Lax. Rev. Mod. Phys. 38 (1966) 541. [9]J.M. Sancho, M. San Miguel. S.L. Katz and J.D. Gunton, Phys.Rev.A26 (1982) 1589. [10] J. Masoliver, B.J. West and K. Lindenberg, Phys. Rev. A 35 (1987) 3086. [11] R.F. Fox, Phys. Rev. A 34 (1986) 4525. [12] J.M. Sancho, F. Sagues and M. San Miguel, Phys. Rev. A 33 (1986) 3399. [13]K. WOdkiewicz, J. Math. Phys. 23(1982) 2179. [l4]P.Grigolini,Phys.Lett.A 119 (1986) 157. [l5]L. Cao, D. Wu and H. Wan, Phys. Lett..A 133 (1988) 476. [16] RH. Terwiel, Physica 74 (1974) 248. [17] K. Lindenberg and B.J. West, Physia A 119 (1983) 485. [l8]N.G.vanKampen,Phys.Rep.24(l976) 171. [19]M. Ku~and K. Wódkiewicz, Phys. Leti. A 90 (1982) 23. [20] A. Schenzle and R. Graham, Phys. Lett. A 98 (1983) 319: P. Jung and H. Risken, Phys. Lett. A 103 (1984) 38. [21]A.P. Prudnikov. Yu.A. Brychkov and 0.1. Marichev, Integrals and series, Vol. III (Nauka, Moscow, 1986). [22] R. Dumcke and H. Spohn, Z. Phys. B 34 (1979) 419. [23]J.Luczka,PhysicaA 149 (1988) 245.