CORRELATION TIME EXPANSION FOR ... - Science Direct

PHYSICS LETTERS

Volume 90A, number 1,2

21June1982

CORRELATION TIME EXPANSION FOR MULTIDIMENSIONAL WEAKLY NON-MARKOVIAN GAUSSIAN PROCESSES H. DEKKER l Department of Chemistry, MassachusettsInstitute of Technology, Cambridge, MA 02139 USA 2 and Physics Laboratory TNO, Den Haag, P.O. Box 96864, The Netherlands 3 Received 15 April 1982

The stochastic Liouville equation is evaluated explicitly through first order in an expansion in terms of correlation times of multiplicative gaussian coloured noise for a general multidimensional weakly non-markovian process by means of Novikov’s theorem. It is explained why the present result is new.

I. Introduction. The application of the original ideas of the historical linear Langevin equation [l] to more general nonlinear stochastic processes has usually been confined to the realm of gaussian white noise sources (e.g. ref. [2]). In that case the generated process is not only markovian, but its fundamental transition probability obeys a Fokker-Planck (or generalized diffusion) equation (e.g. refs. [3-51) in lieu of the full ChapmanKolmogorov (or master) equation (e.g. refs. [6-81). Only rather recently there has been more widespread interest in nonwhite (or coloured) noise sources in generalized Langevin (or stochastic differential) equations (e.g. ref. [9]). In the interesting case of weak colouring a systematic evaluation beyond the markovian limit is feasible in terms of the small correlation times of the noise. In particular, if both the correlation times and the intensity of the fluctuations are small, the cumulant expansion has been shown to be useful [9]. However, in the present note I will restrict myself to gaussian noise and demonstrate the usefulness of Novikov’s gaussian theorem [ lo,1 11, which is relatively unknown to the physics community. The assumption of small noise intensity will not be made. Starting from the actual process (or stochastic differential) equations, a stochastic Liouville equation for the probability density will be derived which, in the general multidimensional case, improves upon a previously published result [ 11,121. The origin of the novelty of the present result is clearly revealed.

2. Formal stochastic Liouville equation. Let the following set of coupled stochastic differential equations ib = &P(X)+ wk(X&(t)

(1)

be given. In general p E [ 1,iVj and k E [ 1,&fl, with N #&f. The tk(t) are taken to be gaussian but nonwhite stochastic variables of zero mean and with correlation functions G,(t) S,(t ‘)) = r/J&

t’) .

(2)

’ On leave of absence from the Physics Laboratory TNO, Den Haag, The Netherlands; sponsored by TN0 and a Netherlands ZWO/NATO Science Fellowship. 2 Address during 1981, August 15-1982, August 15. 3 Permanent address.

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0 1982 North-Holland

PHYSICS LETTERS

Volume 9OA, number 1,2

21June1982

The correlation times rkz are conventionally defined by = idim_ j dt’ (t - t’)rkl(t, Dklrkl 0

t’) ,

(3)

Dkl= /iUI_ j dt’rkl(t, t’) . 0

Considering the continuity equation for the density p(x, t) in the phase space spanned by the xc1 and invoking van Kampen’s lemma [9], which states that E @(x(t) - x)) s P(x, t) represents the probability density for the process x(f), one readily obtains the stochastic Liouville equation

w,

-=__

a

0

av

at

app_

a @,f (&$)8(x(t) axr

(5)

-x)> .

3. Novikov’stheorem and formal time scpansion. By virtue of the gaussian character of the noise, the stochastic Liouville equation (5) can be further evaluated using Novikov’s theorem [IO], Its direct application leads to *I df'

rkl(t,

t')([6XY(t)/6jj(t')]6(X(t)X)> ,

which leaves us with the calculation of unaveraged response functions. In view of the weakness of the colouring, (6) will be expanded systematically in terms of the correlation times rkr. Because of causality Gxv(t)/SSj(t’) E 0 if t’ > t. See also (9). Thus the response function is discontinuous at t’ = t. However, it is properly behaved for t’ t f. Therefore, for t’ < t one sets

sxuw = Wdt’)

32”1’O’(t) + (t’ - tyPp(t)

32

v (4$) I

s

_A!? 6x“(0 dt’rl atI &

I

t $(t’

- t)2mq2’(t)

t ... ,

(7)

(8)

q=O,l,....

At first glance, it seems that one only needs to compute 72”~o)(t) and 32 ‘(‘) I (t) in order to obtain the result for (6) through first order in $1. However, in the sequel I will show that an infinite number of higher order terms in (7) effectively contribute to (6) in order rkz. Fortunately, the infinite series can be resummed in closed form. It is these subtle aspects that apparently have been overlooked so far. 4. Responsefunctions. Instead of calculating the response functions by means of MSR operator formalism [ 121, I prefer here to deal with them in the following simple manner, leading to identical results. Formally integrating the original stochastic equations (1) almost trivially yields x”(t) =x’“(O) + j dt” [a“(x(t”)) + @‘k(x(t”)){k(t”)]

.

(9)

0

Hence

*’ As usual, I assume that no confusion will arise from the use of the same symbol 6 for both the delta distribution and the funa tional derivative.

27

PHYSICS LETTERS

Volume 90A, number 1,2

_I-c --ICl y&

SxP(t)

t

dt,,

a* + atip1

sg,O =*,

axv

axv

+iqx(t’))

21June1982

(10)

)

fl

where I again used causality in the sense that, by the very nature of (9) 6xJ‘(t)/6Sk(t’) 3 0 if t < t’. From (10)

one immediately finds the standard result c&?pLO))(t)= l!+“, .

(11)

Differentiating (10) with respect to t’ gives

aw, “p(t’)

+ --&-

i”(t’)

,

(12)

t’

which, using (1) and (1 l), readily leads to wy(t)

= MP, + KQ&t)

aw,

Mpkqyv-

axv

,

aafl -m..m~” a9

k’

(13)

at+ K”kl z au, ~a9

asp, __ lv, . a9

(14)

This result agrees with [l l] +2. 5. Correlation time expansion. I now come to the crux of the present note. Introducing merely (11) and (13) into (7), and inserting that time expansion into the stochastic Liouville equation (6), one gets the result presented inrefs. [11,12]. However, let us consider %?fl$2)(t)_ In view of (13) for %Zgf’(t), it is easily seen that ‘9r~2)(t) will contain contributions proportional to f&t) and to cl(t){,(t)_ Upon applying Novikov’s theorem this will introduce terms in the stochastic Liouville equation that on one hand (i) involve the second power oft’ - t, but on the other hand (ii) involve either the time derivative of a correlation function or an equal time correlation. Since the latter give essentially rise to an inverse power of the correlation time, the final outcome for these terms will again be of first order in rkr. Evidently, this effect persists through all orders. Fortunately, in order to be correct through first order in rk[ one only has to carry along part of the response function, in the sense that for 4 > 2 C-K/Jid(t)

QP klm

&7-l ($0 kl F-1

= aK”kl

ad

dq-2 r 1(t) ’ &ct) + QPklm 6n(t) dt4-2

awm

,p

m

- -

axh

K’kl ,

(1)

where = indicates that this formula is not complete but suffices to evaluate the stochastic Liouville equation through first order in rk[. In (15) I have anticipated the consideration of stationary noise, such that rkl(t, t’) = ,?k& - t’).

6. Result. Introducing now (1 l), (13) and (15) into (7), and inserting the expansion into (6), I obtain

*2 In the notation of ref. [ 111, (~0-f Vp, dv -+g,gv,Mfiv -+Msv, and Kppv -+ KgvM.

28

(16)

Volume90A, number 1,2

PHYSICSLETTERS

@(x9t)_ -&pP+D

-!?

a+

:a’

8”

a9 t W2)

(t?“, - rklMvI)P

.rdt’r,,(t, t’) [tt’(

+ (1/2!)(t’ - t)2d{,

t&(t)

/dt +

0

dt’ rk,(t, t’) ([(l/z!)(t’

- t)2s,(t)

+ (l/3!)@

- t)3dr,/dt

,

where rkl and Dkl, defined in (3) and (4), may stih depend on time for finite t. The be elegantly resummed to a closed from, with the result

a @ptD

ap(x9t) -=-_ at

a

a 9& axr

+ $,,

“k

a

81

k1G

aiv

--

a (“I-

k G

-x)>

(17) occurring infinite series can

-x)1

Qvlmn j dt’ rkl(t, t’) 1 dt” j dt”’

o

QQ

?-

k a9

-!-

k a9

t

+a IV’ &vlm ae kaxv ta

6”

k1 ae

21 June 1982

-x)>

t 0(r2) ,

(18)

which clearly shows the differences with refs. [ 11,121. Finally, once more applying the gaussian theorem, using the response function (11) through zeroth order in rkl and somewhat rearranging terms, one finds

ap(x,t, at

tvklmn(t)

a #p+ D

ae

a&’ pk -% 6” axv

k

n

-! axi

aXy

(byI - @!f”~)P + ekl,&t)

K”,,Pt

CMa’,

OK",,,,

axv

a 6”,p

axA

0(r2),

(19)

where Ekrmn(t) Z 1 dr &(r) 0

1 d7’ I-” 0

dr” q,&“)

,

w-0

0

(21)

7. Di~cr.&on. Let me in this brief discussion apply (19) to the case of uncorrelated noise sources of unit intensity, i.e. Fkr(r> 3 Sk&(r) , and neglect transients on the scale Of the rk[. with Dkl = Sk, and rkl = $iikl, (19) reduces to

am,0 =_ at

-?-_lYPpta,,

aiv k

aiw

k

$

tavk- $M”k)P

&,v

axv

where ekl= ekk&=) and qk[ E nkkl++

-?- IV’ -f!- iP’ -!axI k axv Iad

Khk,P

t O(T~),

If, in addition, I insert for Fk(r) the Uhlenbeck-Omstein

(22) form [9,14] 29

PHYSICS LETTERS

Volume YOA, number 1,2

lTk(T) = (l/Qe-

‘+k

)

21June1982

(23)

then Ekl=rk,

nk[ = #k

+ 7$ .

(24)

Such a result could never arise from the formula given in ref. [ 11,121. The new terms on the rhs of (22) persist even if all rk are equal *3. In conclusion: I have shown explicitly - by example in (24) - that the present result (19) indeed is of order rk, and in what sense and why it corrects an earlier reported version. I would like to thank Prof. Dr. 1. Oppenheim and Dr. M. Pagitsas for their interest. A portion of this work was supported by the National Science Foundation under grant # CHE-79-23235. *3 As in refs. [ 11,121, it is worth noting that in that case the third order derivatives of P in (19) and (22) vanish by virtue of the antisymmetry property KCCkl= -Kfilk. For nonequal rk, aB additional terms involving Kpkl vanish identically in phase spaces with zero riemannian curvature (e.g. [ 15I), because K+f can be expressed in terms of phase space anholonomity wfihe [S] as K@kl= SxkSul (ww,~ - wr&). In globally flat spaces w@,I,~= 0 and any multiplicative noise can always be transformed into additive noise by means of an appropriate coordinate transformation.

References [l] [2] [ 31 [4] [5] [ 61 [7] [8]

[Y] [lo] [ll] [ 121 [13] [ 141 [15]

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P. Langevin, C.R. Acad. Sci. 146 (1908) 530. L. Arnold, Stochastic differential equations (Wiley, New York, 1974). R.L. Stratonovich, Topics in the theory of random noise, Vol. I (Gordon and Breach, New York, 1963). H. Haken, Rev. Mod. Phys. 47 (1975) 67. R. Graham, Z. Phys. B26 (1977) 397. I. Oppenheim, K.E. Shuler and G.H. Weiss, Stochastic processes in chemical physics: the master equation (MIT Press, Cambridge, MA, 19 77). N.G. van Kampen, Stochasticprocesses in physics and chemistry (North-Holland, Amsterdam, 1981). H. Dekker, Physica 103A (1980) 55,80. N.G. van Kampen, Phys. Rep. 24C (1976) 171. E.A. Novikov, Sov. Phys. JETP 20 (1965) 1290. M. San Miguel and J.M. Sancho, Phys. Lett. 76A (1980) 97. M. San Miguel and J.M. Sancho, in: Stochastic nonlinear systems in physics, chemistry and biology, Synergetic series Vol. 8, eds. L. Arnold and R. Lefever (Springer, Berlin, 1981) p. 137. P.C. Martin, E.D. Siggia and H.A. Rose, Phys. Rev. A8 (1973) 423. G.E. Uhlenbeck and L.S. Ornstein, Phys. Rev. 36 (1930) 823. H. Dekker, Physica 103A (1980) 586.