Stochastic dynamics for systems driven by correlated noises

31 January 1994 PHYSICS LETTERS A

Physics Letters A 185 (1994) 59-64

EI~qEVIER

Stochastic dynamics for systems driven by correlated noises Li Cao, Da-jin Wu CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China, National Laboratory of Laser Technology, Huazhong University of Science and Technology, Wuhan 430074, China t Received 21 September 1993; revised manuscript received 2 December 1993; accepted for publication 6 December 1993 Communicated by A.R. Bishop

Abstract Recently, Fulinski and Telejko [ Phys. Lett. A 152 ( 1991 ) 11 ] have studied the effect of correlation of additive and multiplicative noises. However, they have not obtained a general Fokker-Planck equation (GFPE) which can describe the dynamics of the system driven by multi-noises with an arbitrary degr e of correlation. In this Letter, we have derived a GFPE for a onedimensional system from the Langevin equation (LE) with multi-noises and an arbitrary degree of correlation between these noises. Using this GFPE, we study the single bistable kinetic process driven by correlative additive and multiplicative noises. In conclusion, the results obtained in this Letter provide a correct foundation for the treatment of the effect of correlation of the noises.

1. Introduction On the level of the Langevin-type description of dynamical systems, the presence of correlation between the noises changes the dynamics of the system [ 1 ]. The G F P E corresponding to the LE with correlated noises has not been obtained in Ref. [ 1 ]. In this Letter, we derive a G F P E for a one-dimensional system from the LE with correlated noises [2 ]. By virtue of the GFPE, we study the single bistable kinetic process driven by the additive and multiplicative noises with an arbitrary degree of correlation. Several conclusions are stated in Section 4.

2. General Fokker-Planck equation for a one-dimensional system driven by correlated noises The general case of the o n e - d i m e n s i o n a l LE with correlated noises has the form ~ = h ( q , t ) + ~ gj(q, t ) ~ . ( t ) .

(1)

J

Here ~ ( t ) are Gaussian white noises given by ( ~ ( t ) ) = 0 and

( ~ ( t ) ~ ( t') ) =2D/~( t - t ' )

,

Mailingaddress. 0375-9601/94/$07.00 © 1994 Elsevier ScienceB.V. All rights reserved SSDI 0375-9601 ( 93 ) E0983-H

(2a)

60

L. ('ao, D. ~Vu /Ph.vstcs Letters,l 185 (1994) 5 9 - 6 4

( ~,( t )~,( t') ) = 2)tj,( D,D, ) ~/zis( t - t ' ) .

( 2b )

In (2b) the parameter 2# measures the strength of correlation between the noises, Now, we wish to get a FPE corresponding to Eqs. (1) and (2). To this end, we start with the definition of drifi and diffusion coefficients [3], D~l)(_v, t ) = lira
!3~

and D(2)(.v, t ) = ½ l i m ~ ( ( a q ) : ) r ~0

T

where {+ r

l+r

f

d,'+ f

I51

!

in which x denotes the peaked value of the stochastic variable q( l ), i.e., P( q, t ) = ~( q - .v). FoF clarity, we first consider the stochastic integral in Eq. ( 5 ) as a Stratonovich integral [2 ] ~. The llo case can be considered in the same way without difficulty. In the following we assume the formal expansion

h[q(t'), t'] =h(x, t')+h'(x, t') &[q(t ),t']=&(x,t')

[q(t')-x]

+ ....

gj(x,t')[q(t')-_v]+ ....

(6)

Here

h'(x,

t')=

O h ( x , t')~ ~qh[q(t'), •

O

g)(x,t')=

['] q(t'

0 g/(x,t')-~ ~q&[q(t'),

t'] ql,'l=,

171

Using the expansion (6) we get from Eq. ( 1 ) /+r

t+r

j t I+T

+ y. f g:(x,t'l~j(t')dt'+

t t+r

y. ~ g',(x.t')[q(t')-.vl~,(t')dt'+

....

(8)

t

B y iterating the quantities q ( t' ) - x in Eq. (8) and taking the average with respect to the noise distributions, we have ~ In Ref. [ 2 ], Fox uses ordered operator cumulants and van Kampen's lemma to obtain a correlation time expansion for colored nomc,

stochastic processes. In the limit of the correlationtime going 1o zero, and for a singlevariable, Fox's equation (25) gives the Stratonovich version of our equation ( 13 ).

L. Cao,D. Wu/ PhysicsLettersA 185(1994)59-64 t+z

t+z

t t+r

t'

t

t

t'

t+z

+ ( f h'(x,t')(~ ~gj(x,t"),j(t")dt")dt'l+...+(~ t

f gj(x,t'),j(t')dt' I

t t+z

61

t

l'

+(~ f g,(x,t') fh(x,t")dt"~(t')dt) l

t

t+r

t'

+(~ f gj(x,t')(~ fg~(x,t")~(t")dt")~(t')dt')+ .... l

(9)

t

In the limit of r--, 0, Eq. (9) leads to the general expression of the drift coefficient,

D"(x,

t ) = lim ( A q ) z~O

=h(x, t)+ ~Djgj(x, t)gj(x, t)+ ~ 2j~(DjDi)'/2gj(x, t)g,(x, t)

"C

j

j>i

+ ~ 2j~(DjD,)~/2gj(x, t)g,(x, t) .

(10)

j
To get Eq. ( 10 ) from Eq. (9), the statistical properties of Gaussian white noises ~j(t) ( Eq. (2) ) have been used. Similarly, using Eq. (8) the general diffusion coefficient reads

D~2)(x, t ) = ~

lim ( ( A q ) 2 ) - ~ Z~oo

"C

j

Djg~(x, t ) + 2

~

2ji(DjDi)~/Zgj(x, t)g,(x, t) .

(11)

j> i

When the stochastic integral in Eq. (5) is considered as the Ito integral, it can be shown that the drift coefficient is D~I)(x,

t) =h(x, t)

(12)

and the diffusion coefficient is the same as in the Stratonovich case. Therefore, we obtain a GFPE corresponding to LE ( 1 ) with (2),

O p(x,

O D~l)(x, t)P(x, t)+ OOZ D~2)(x, t)P(x, t) X 2

t)= -

(13)

where

D~I)(x, t) =h(x, t) (14a) •

j>i

D~E)(x, t)= ~ Djg}(x, t)+2 ~ 2ji(DjD,)l/Zgj(x, t)gi(x, t). j

j
(14b)

j>i

In Eq. (14), v = 1 for the Stratonovich calculus and v = 0 for the Ito calculus, respectively. As it has turned out, the coefficients Dtn) (x, t) vanish for n~> 3 for Eq. ( 1 ) with Gaussian white noises 5.

L. Cao, D. Wu / Physics LettersA 185 (1994) 59-64

62

3. The single bistable kinetic process driven by correlative additive and multiplicative noises We use the general formulas obtained in Section 2 for an important system, the single bistable system driven by correlative additive and multiplicative Gaussian white noises,

~t=-aq-bq3-q~(t)+~l(l)

(h>0)

(15)

with

( ~(t) ) = ( q ( t ) ) = O .

(16a)

( ~(t)~(t') ) = 2 D ~ ( t - t ' )

(16b)

(q(t)~l(t'))=2oe~(t-t'),

t l6cl

( q ( t ) ~ ( t ' ) ) = ( ~ ( l ) q ( t ' ) ) = 2 2 ( o ~ D ) ~ / ' ~ a ( l - r ')

()~2~<1),

(16d)

This is a special case of Eqs. ( 1 ) and (2) with gt=l,

g2=-q,

Dl=oe,

D2=D,

-t. 12 = Z 2 1

=-~

.

(17~

Therefore, we get a FPE of system (15) and (16) by substituting Eq. (17) into Eq. (14). This leads to thc following expressions of the drift and diffusion coefficients of system ( 15 ) and ( 16 ),

D ( l ~ ( x ) = - b x 3 - ( a - v D ) x - v).(olD) ~2

i 18

D(23(x)=Dx2+oz-2).(oLD)I/2A .

(19

and

By means of results (18) and (19), the s t a t i o n a ~ distribution P~,(.v) can be obtaincd from the formula

P~t(x)=N~exp

D~l)(X')dx' D{~(.v, )

.

(20

We do not give the expression of P~dx) here. but rather give two important results of Eqs. ( 18 ) and ( 19 ( i ) The algebraic equation which governs the stationar3 ~most probable values Vm o f a . hx 3 + [ a + ( 2 - v ) D ] x m - ( 2 - v)2(eeD)~/e=O.

(21)

(ii) The expression of the critical parameter a~ at which the transition between the unimodal ( a > a ~ ) and the bimodal (a < ac ) distribution occurs,

a,. = - ( 2 - v ) D - 3b[ ( 2 - v)2)~2ogD/4b 2 ] l~a

( 22 )

4. Conclusions (i) In Ref. [ 1 ], a FPE corresponding to Eqs. 15 ) and (16) (i.e., Eqs. (4) and ( 5 i of Ref. [ 1 ] ) has been written based on the assumption that the relation t/d/=dwl,

~dt=(l-)t)dw2+)~(~72/al)dw,,

(231

(cr~ is the strength of dw~ and a2 the strength of dw2) is correct, where ~,,~ and we are uncorrelated Wiener processes. However, relation (23) is incorrect. The reasons are the following. (a) It leads to incorrect drift and diffusion coefficients (Eq. (8) of Ref. [ 1 ] ) and then the following incorrect results: the algebraic equation which governs the stationary most probable values Xm o f x (Eq. (10) of Ref. [ I ] ),

L. Cao, D. Wu / Physics Letters A 185 (1994) 59-64

63

bx~ + [ a + ( 2 - u ) ( 2 2 2 - 2 2 + 1 ) D l X m - ( 2 - v)2(aD)'/2=O,

(24)

and the expression of the critical parameter a¢ (Eq. ( 11 ) of Ref. [ 1 ] ), a¢ = ( 2 - ~) ( 2 2 - 2 2 2 - 1 ) D - 3b[ ( 2 - u)z22aD/4b 2] 1/3

(25)

This may be seen from the fact that when the additive noise disappears ( a = 0), in other words, the system is driven by one noise alone, the parameter 2 characterizing the correlation between the noises still remains in Eqs. ( 8 ), ( 10 ) and ( 11 ) of Ref. [ 1 ]. (b) The relation (23 ) is incompatible #2with the required statistical properties for q(t), ~(t) (Eq. (5) ofRef. [ 1 ] ) and for dwl, dwz (Eq. (2) ofRef. [ 1 ] ) except that 2 = 0 a n d 2 = 1. Therefore, the FPE (7) and (8) of Ref. [ 1 ] is equivalent to the LE (4) and (5) of Ref. [ 1 ] only at the values 2 = 0 and 2=1. (ii) The GFPE ( 13 ) and (14) is obtained in a rather direct way, that is, it is obtained based on the LE ( 1 ) and (2) and the definition of drift and diffusion coefficients. Therefore, our results ( 18 ) - ( 2 2 ) are correct for any values of 2 and contain the 2-dependent terms only in the form 2 (otD) ]/2. (iii) It must be pointed out that the results ( 1 8 ) - ( 2 2 ) provide a correct foundation to study the effects of correlation of the noises quantitatively, and the stochastic parameters o~, D, v, and 2 can be determined by measurement of the critical parameter ac and the stationary most probable values Xm of X. Based on the results ( 18 ) - ( 22 ), the following conclusions can be drawn: -0, 3.0

1

2.5

2 2.0 3 1.5 4

1.o

o,5

I

0,0

0.0

O..S

I

,

1.0

Fig. I. The critical parameter ac as a function of the degree of correlation between the multiplicative and additive noises 2. The value of the parameters are: b = l and D = 0 . 5 . Curves ( 1 ) and (3) are for a = 0 . 5 and curves (2) and (4) are for a = 0 . 1 . Curves (1) and (2) correspond to v = 0 (the Ito interpretation of Langevin equation ( 1 ) ) and curves (3) and (4) correspond to v = 1 (the Stratonovich interpretation of Langevin equation ( 1 ) ). #2 It can be seen that the combination of relation (23) with the required statistical properties for ~/(t), ~(t) (Eq. (5) of Ref. [ 1 ] ) and for dWl, dw2 (Eq. (2) of Ref. [ 1 ] ) leads to the following algebraic equation for 2 : 2 - 2 = 0 . This equation limits the values of 2 to 2 = 0 and 2 = 1.

64

L. Cao. D. Wu /Physics Letters A 185 (1994) 59-04

( A ) The case of only one noise. It can be seen from (21) and (22) that when the multiplicative noise is the only noise (i.e., D:/: 0 and c~= 0), the following feature can be shown. The algebraic equation that governs the stationary most probable values Xm o f x (Eq. (21) ) becomes bx 3 + [ a + ( 2 - u ) D ] x m = 0 , it obviously differs from the deterministic case bx3m-t-aXm = 0 by the term ( 2 - P ) D X m . This term shifts the couple o f Xm from +_ ( - a / b ) ~/2 (the deterministic case) to _+ { [ a + ( 2 - v ) D ] / - b } ~/2. The critical p a r a m e t e r ac makes an interesting change in that the value o f ac = 0 for the deterministic case shifts to the value - ( 2 - v)D. For example, if the LE ( 1 ) o f the p r o b l e m considered obeys the Ito interpretation, we have a c = - 2D, whereas if it obeys the Stratonovich interpretation, ac = - D. W h e n the a d d i t i v e noise is the only noise (i.e., c~ ~ 0 and D = 0 ), it cannot affect the results o f x m and a,,. That is, Eqs. (21 ) and (22) read bx3~ + aXm = 0 and ac = 0, as in the deterministic case. The a d d i t i v e noise only gives an effect on the diffusion coefficient D (2 ~ (x, ! ) as usual. (B) The case o f simultaneous presence o f both a d d i t i v e and multiplicative noises. If they are uncorrelated ( 2 = 0 ) , we see from Eqs. (21) and (22) that Xm and a~ do not differ from the case o f the presence o f multiplicative noise alone except for the diffusion coefficient (see Eq. ( 1 9 ) ). However, the presence o f the correlation between the noises (2 4: 0) causes the additive noise to play an i m p o r t a n t role in the bistable properties o f the system as can be seen from Eqs. ( 1 8 ) - ( 2 2 ) . To see the influence o f the coupling p a r a m e t e r 2 on the critical p a r a m e t e r a~, the d e p e n d e n c e o f ac on 2 and c~ is plotted. It is interesting to point out that Fig. 1 exhibits a sensitive d e p e n d e n c e ofa~ on 2 in the region o f small 2. Fig. 1 also shows the role of a d d i t i v e noise on the value o f a~ when 2 ~ 0. ( C ) The role of the macroscopic p a r a m e t e r b. F r o m Eq. (22) we see that the p a r a m e t e r b, describing the nonlinearity o f the system, affects the critical p a r a m e t e r a, only when there is a correlation between the noises (2 ¢ 0). This gives the possibility to d e t e r m i n e the stochastic parameters c< D, v and 2 by means o f the measurement ofa~ and Xm as the function o f both macroscopic p a r a m e t e r s a and b as pointed out in Ref. [ 1 ].

Acknowledgement This research was supported by the National Science Foundation of China.

References [ 1] A. Fulinski and T. Telejko, Phys. Lett. A 152 ( 1991 ) 11. [2] R.F. Fox, Phys. Len. A 94 (1983) 281. [ 3 ] H. Risken, The Fokker-Planck equation (Springer, Berlin, 19841.