non-adiabatic elimination of variables in stochastic ...

International Journal of Modern Physics B, VoL 9, No.6 (1995) 679-694

© World Scientific Publishing Company

NON-ADIABATIC ELIMINATION OF VARIABLES

IN STOCHASTIC PROCESSES BY MEANS OF PATH INTEGRAL

AND INFLUENCE FUNCTIONAL METHODS

HORACIO S. WIO* Centro Atomico Bariloche, Comision Nacional de Energ(a Atomica

and

Instituto Balseiro, Universidad Nacional de Cuyo 8400 C. de Bariloche, Argentina

s.

C. BUDDE and C. BRIOZZO Facultad de Matematicas, Astronom(a F(sica Universidad Nacional de Cordoba 5000 Cordoba, Argentina and P. COLET Departament de F(sica, Facultat de Ciencies, Univeraitat de lea Illes Balears,

E - 01011 Palma de Mallorca, Spain

Received 13 June 1994 We present a novel scheme for the non-adiabatic elimination of variables in stochastic processes, based on a path integral representation of the probability density and the use of an influence functional. We analyze in particular the case of multivariate Fokker­ Planck equations, or equivalently a set of coupled Langevin equations driven by white noises, and discuss some examples where exact or approximate results are obtained.

1. Introduction

A quite remarkable feature of complicated dynamical systems is the possibility of having reduced causal descriptions. This means: to have a description in terms of a reduced set of variables, much smaller than the set including all degrees of freedom of the system. This fact is usually explained in terms of the existence of an interplay of mechanisms that evolve on quite different time scales. The equations of motion of such systems can often be simplified by eliminating the fast variations. This procedure is the prototype of all adiabatic elimination procedures which has been used as the basis of Haken's slaving principle. 1 The basic physical assumption is that when such time scales are well separated, with some of the variables varying over a time scale very much shorter than that of the rest, and if dissipation is present, *Member of CONICET, ARGENTINA. 679

680

H. S. Wio et al.

it is possible for the fast variables to relax to a quasistationary state, in which their values follow the values of the slow variables. This problem has recently been addressed by several authors. 2 ,3 However, in those cases where the above mentioned separation in time scales is not strictly fulfilled, such an elimination procedure is not so obvious. On the other hand, in the context of the kinetic or transport description of physical and chemical phenomena, there is a wide range of situations where it is desirable or even necessary to describe the problem not in terms of a full set of macroscopic variables, but only via a small (and sometimes ad hoc) set of relevant variables. A few examples include the kinetics of multicomponent chemical reactions, cluster growth in homogeneous nucleation, kinetics of phase transitions, heavy ion reactions at low energies, etc. 4 In this work we present a novel approach for eliminating irrelevant variables inspired on Feynman's infiuence functional method. 5 Because this approach does not resort to adiabatic arguments, it is able to eliminate fast as well as slow vari­ ables. The starting point of our presentation is the assumption that the complete description, in terms of the complete set of macroscopic variables, is Markovian. This assumption is not strictly necessary, but simplifies the argument of having a path integral representation of the conditional probability. For instance, such a solution can be obtained by a reiterative use of the Chapman-Kolmogorov relation, employing some adequate representation of the short time propagator. Here we discuss the case of multivariate Fokker-Planck equations, or, equivalently, a set of coupled Langevin equations with (additive or multiplicative) white noise sources. Indeed, the case of colored noise can also be handled. 6 We have put the emphasis on the form of the solution of the problem, i.e. the reduced transition probability after the elimination of variables has been carried out. The finding of the form of the reduced (and foresight non-Markovian) equation is not addressed here and will be discussed elsewhere. The paper proceeds as follows. The method is presented in Sec. 2. It is firstly formulated in configuration space, where a Ansatz for the influence functional is introduced. Such Ansatz is justified in a weak interaction limit, and a physically sound interpretation is assigned to it. Afterwards, we make the formulation in phase space, with its corresponding Ansatz. Section 3 is devoted to discussing a set of examples where we have evaluated, in an exact or in an approximate form, the influence functional· and show that it adopts forms coincident with the proposed Ansatz. Such examples include: a simple model corresponding to an oscillator in phase space, the colored noise problem, and the case of nonlinear noise. The first of these examples shows the possibilities of the present method for doing a non­ adiabatic elimination of variables. In Sec. 4 we present a final discussion. 2. The Approach

2.1. Configuration space representation A preliminary account of the results in configuration space was given in Ref. 7. As was done there we will follow the notation and use some of the results of Refs. 8

Non-Adiabatic Elimination of Variables in Stochastic Processes. . .

681

and 9. 'vVe start by considering a multivariate Fokker-Planck equation that, using a short hand notation, reads

o

at per, tiro, to) =

0

- or (h(r)P(r, tiro, to))

o

0

+ ar]]) orP(r, tiro, to)

(Ia)

where: r = (rI' r2, ... , rN), her) (hI(r), ... , hN(r)), %r (%rb ... , 0/ or N ) j P( r, tiro, to) is the conditional probability for reaching the point r at time t, if we started from ro at to; and]]) is the diffusion matrix with components (]]))ij = Dij, which we assume to be non-singular. The set of equivalent Langevin equations can be written as (Ib) with

~k(t)

white noises of zero mean and correlation functions (Ie)

Following Refs. 8 and 9, we arrive at the full propagator (by using a middle point discretization prescription)

where D[r] is the path-integration measure as usual. \Ve now separate the whole set of variables into a subset of relevant variables {x}, and a subset of the remaining (irrelevant) variables {y}, according to

{r} = {{x}, {y}}.

(3a)

{her)} = {{hx(r)}, {hy(r)}}

(3b)

We also write and also adopt the separation hx(r)

h o , x(x)

hy(r) = ho,y(Y)

+ hI, x(x, y)

+ hI,y(x, y).

(3c)

In order to simplify the notation in what follows we will consider only one relevant and one irrelevant variable: x and y respectively. The extension to several relevant and irrelevant variables is straightforward. The full Lagrangian

err,o r] = 2'1

[0r -

her) ]t

I 0 r-h(r) +2'orh(r) 0

[

]

(4)

682

H. S. Wio et al.

can be written as the sum of three contributions (5)

which are given by

[x, x] =

Cx

containing all the dependence on x and

Cy[iI, y] = same with y and Cindx,

X,

x alone.

y alone.

y, y] = including all the interaction terms.

The definition of C x and C y implies the possibility that some of them (or both) could not be bona fide Fokker-Planck Lagrangians. That means that they would not have the structure indicated in Eq. (4). We now define the marginal or inclusive conditional probability by means of

where we have taken a weighted average on the initial distribution of the subset {yo} (with the simplifying, but not necessary, assumption that the initial distribution at time to, is separable, i.e.: Po(ro, to) = (xo)Py(Yo)), and integrated over all possible final values {Yj }, as usual. In order to make the connection with the Feynman approach, the marginal probability can be written as D ( Xj, t t) j i .Linc xo, 0

I 2J j-r.[x v 1e- / :: dt.cx[x,~lF[x,

XO]

(7)

where the functional F[x, x] is given by (8a) Due to the similarity with Feynman's influence functional,5 we will call this quantity the stochastic influence functional. Io It is clear that functional defined in Eq. (8a) shall only have some, and not all, of the properties of Feynman's influence functional,5 as we have indicated in Appendix A. In the same spirit of Feynman's scheme, we write

F[x,

xl =

exp

(1:'

dt,8[x(t), X(t)])

(8b)

and introduce the following Ansatz for the stochastic influence exponent ,8[x, x] (at variance with Feynman's case, here it is not a phase!)

,8[x(t), x(t)] = fo(t)x(t)

t dt'x(t)C(t, t')x(t')

ito

(ga)

Non-Adiabatic Elimination of Variables in Stochastic Processes. • .

683

where fo(t) will have the form fo(t)x(t)

= fOO(xf, xo)8(t

to) +

t

lto

dt'fo,1(t, t')x(t') ,

(9b)

foo being some function of the initial and final coordinates Xo and x f. We will show that such an Ansatz can be justified, and that a sound interpretation can be assigned to it, for a broad class of situations. In order to see how such a result could arise in a practical case we will assume the simplified (linearly coupled) situation ho,x(x)

= Ax;

= sq(y);

h1,x(x, y)

h1,y(x, y) = sBx;

ho,y(y)

= ~(y)

(lOa)

DTI,

Jl)l

]: being the identity matrix. Then the interaction Lagrangian will become 'cint

=~

(Aq(y)

+B~(y)

By)x(t) - q(y)x(t»)

= s(J1,(y, y)x(t) - ¢(y)x(t» ,

(lOb)

J1,(y, y) being a function of y and y, and ¢(y) is a function of y only. Replacing (lOb) in (8a) we obtain F[x, x] =

J J dYf

dyoP(yo)

J

t

V[y]e- ft :

dt{.cll[Y,YJ+e[IL~-4>x]} .

(11)

In the case of weak coupling (s « 1), we can expand the second exponential in powers of the coupling constant s and perform the path integral, obtaining F[x,

J JJ

xl = Fo (1 + s2

s

dt

dt«((J1,(Y, y)}}x(t) - ((¢(y)}}x(t» dt'(x(t) ((J1,(y, Y)J1,(y, y)))x(t')

+ x(t)((J1,(y, y)¢(y»))x(t')

+ x(t) ((¢(y)J1,(y,

y)}}x(t')

+ x(t)((¢(y)¢(y»))x(t')) + ...

(12)

where the angular brackets denote an average defined through ((B(y(t»)) = F01

J J dYf

dyoP(yo)

J

V[y]8(y)e-

f:: dt.c,,[y,Y].

(13)

Fo is the normalization factor. This factor will be one only if 'cy is a bona fide Fokker-Planck Lagrangian.

684

H. S. Wio et al.

Transforming the perturbative expansion (12) into a cumulant expansion, and restricting ourselves to the case in which second order perturbation leads to reason­ able accuracy (corresponding to a Gaussian approximation), the influence functional becomes

F[x, xl

Jdt(~o(t)x(t) J Jdt'(X(t)~l(t,

= .1'0 exp ( +

-

g

- o(t)x(t»

t')x(t') + ...

dt

+ X(t)l(t, tl)xCt'») , with

~o(t) = ((~(y(t), y(t»)};

Jtl(t, t') l(t, t')

(14a)

oCt)

= (((y(t»))

((~(y(t), y(t»~(y(t'), y(t'»}};

= ({(y(t»(y(t'»));

(14b)

etc.

x,

we see that (14a) and (b) After integrating by parts the terms depending on acquire the desired structure indicated in Eq. (9). What we intend to do next is to argue in order to reach a meaningful inter­ pretation of the above Ansatz. We return to Eqs. (7) and (8b) and ask about the stationary path for the action associated with the effective Lagrangian. Such a path is given by the solution of the Euler-Lagrange equation

d0 - -.ex:::: 0 (d--/3 0 - -/3 0) --ex dt ox ox dt ox ox

0

== -Sl[x, xl

(15)

which can be rewritten as (v = x) o

(16)

v=

It is now necessary to calculate or to make reasonable models for the influ­ ence (correspondingly for Sl[x, x]). In order to give a plausible interpretation of the Ansatz (9) or the result (14), we will profit from considering the underlying stochas­ ticity of the processes at hand. As was done in different contexts l l we assume that Sl[x, xl can be written as

-J

dt'a(t - t')x(t')

+ fIlet)

(17)

where the first term would correspond to a systematic known effect, and if I is a random function of which we only know its probability distribution functional PUII(t)]. Then in order to get sensible results we must consider the average of the

Non-Adiabatic Elimination of Variables in Stochastic Processes. . .

685

influence functional leading to the form (17), over such a distribution, Le. (lSa) When P[fjd is a Gaussian distribution of zero mean, such an average gives the result 5

F"v[x,

xl

exp (-

J

dtfo(t)x(t) -

JJ dt

dt'x(t)C(t, t')x(t

l ))

(ISb)

where C(t, t') (fjl(t)!tl(t')) is the correlation function of the stochastic force. Then we find that the proposed Ansatz can be interpreted as the effect of a Gaussian colored noise, produced by the eliminated variables, on the relevant ones. If the correlation function C(t, t') (or aCt t') as one must be proportional to the other if a fluctuation-dissipation relation is valid) has a very short correlation time (short memory), the systematic contribution becomes

and the above effect would correspond to a Gaussian white noise.

2.2. Phase space representation We can generalize the previous situation by including in Eqs. (1) the possibility of multiplicative noises. Then we write Eq. (lc) as (19)

where as before the 6(t) are white noises with zero mean and correlation function given by Eq. (ld). As before in the configuration space, we can write a phase space representation of the transition probability P(rjtjlroto), in a middle point discretization,6b,g as

P(rj, tjlro, to)

=

J J x

V[r]

V[p]e

e-(1/2)

I,~

dt

I:; dt{iP[~-h(r)1+(1/2)

I:; dt'p(t)C(t, t')p(t

}

l )

(20)

,

p being the conjugate momentum to r, and C the noise correlation matrix. We also adopt the separation (3a, b, c) and, as before, we reduce the problem to only two variables x and y. Now we repeat the margination done in Eq. (6), that is we integrate over the initial distribution of y: P(Yo), and over its final values Yj. As in Eq. (7) now we have xl+(1/2)dho. ",/dx} '!

x e -(1/2) f '0

,

dt 1'0

t')Px(t') F[x,

Px]

(21)

686

H. S. Wio et 0,1.

where we have introduced the concept of stochastic-phase-space influence functional through

x e -1/2 I:t

dt

1:

0

dt' py (t)cyy(t, t' )py(t')

(22a)

where the interaction contribution to the action is given by

(22b) Now, we introduce the Ansatz for the phase-space influence functional, equivalent to the one we have proposed in configuration space

r[X, Px 1-- e - It~ dt{iP x h(x)+H(x)}-(1/2) I:t dt I: dt'Pxc(t, t')px o .

oF

(22c)

In order to justify such an Ansatz, we adopt the same simplified situation as in Eq. (lOa). In the present case it also implies that C xy = C yx = O. Then, the action in Eq. (22b) reduces to (22d) Considering again the case of weak coupling (t:

F[x, Px]

Fo

+

«

1), we can expand and obtain

(1 - Jdt(Px(t)((g(y(t)))) + ((Py(t)}}Bx(t)) it:

JJ dt

dt'(Px(t){{g(y(t»g(y(t'»)})Px(t')

+ Px (t)( (g(y(t) )py(t'») }Bx(t') + Bx(t){{py(t)g(y(t'»))Px(t') + B 2 X(t)((py(t)py(t »))x(t')) + ... l

where the angular brackets indicate the following average:

(23a)

Non-Adiabatic Elimination of Variables in Stochastic Processes. . .

687

Fa, as in Eq. (13), is the normalization factor, and S(y) indicates all the terms in the exponent of Eq. (20a) that depend on y or Py only. Again, transforming the perturbative expansion into a cumulant one, we can reach the form of Eq. (22c) with the following identification hex) = ((g(y(t))))

+

J

dtIB((g(y(t))py(tl)))X(t ' ) +

c(t, t') H(x) = iB{{py(t)))x(t)

J

dt'Bx(t)((py(t)g(y(t'))))

((g(y(t))g(y(t ' ))))

+

(24)

J

dt'B'2x(t)((Py(t)Py(t ' )))x(t').

In this form, we see how it is possible to justify the structure of the Ansatz (22c). The interpretation in this case will clearly be coincident with the one found in the configuration space representation. We would like to remark here that both Ansatz, and more in general, the scheme of writing an influence functional, will work in every discretization,8,g if we adopt the adequate changes.

3. Some Examples In this section we discuss a few examples where we have, in exact or approximate form, evaluated the influence functional.

3.1. Simple model This model, which resembles a simplified version of the Lotka-Volterra model, corre­ sponds to an oscillator in phase space, where both the coordinate and the momenta are subject to external independent noises. It is described by the set of equations

x= by + 6(t) o

y

(25a)

-bx + 6(t)

where ~j(t) are white noises of zero mean and correlation function (~j(t)~k(t')) = 2DDikD(t - t'). Equivalently, the associate Fokker-Planck equation for the proba­ bility density P(x, y, t) is

8 8 8t P (x, y, t) = - 8x [byP(x, y, t)]

8

+ 8y [bxP(x,

2

8 ) + D ( 8x 2 + 8y2 P(x, y, t) . fj2

y, t)]

(25b)

According to Eq. (2) it is easy to get the form for the transition probability in a configuration space path integral representation. As we can see, the involved Lagrangian will be quadratic in x, y, and and then we can do the path integra­ tion in an exact way. If we do the margination processes in order to eliminate the

x y

688

H, S, Wio et a/.

variable y, the exact expression for the inffuence functional is (assuming the initial distribution P(yo) = 8(yo), and to = 0) .F[x,

xl ~ exp (zbsinh (b:)COSh (bt) (sinh 2(bt)x(t)2

lot dt'x(t') cosh(bt' ) - 4b lot dt' lot dt"x(t')x(t") (Zb sinh(b(t - tf)) cosh(bt") - 4bsinh(bt)

+ 8( t '

- t")(sinh (b( t

t')) cosh(b( t

til)

sinh(b(Zt - t')) COSh(btll ) ) ) )

(Z6)

which coincides with the Ansatz given in Eqs. (ga) and (b), with a complicated structure for the correlation function. Here it is worth making the following remark. In this example even though both variables (clearly) evolve on the same time scales, thereby making an adia­ batic elimination procedure not applicable, the present approach allows us to obtain the reduced probability distribution. Also, we do not need to resort to the pertur­ bational expansions in a coupling parameter. Incidentally, in this case it is also possible to obtain the final form of the marginal propagator, as well as the form of the (nonMarkovian) equation that governs its evolution. Both results are shown in Appendix B. 3.2. Colored noise case

We will consider the case of additive Ornstein-Uhlenbeck noise,12 which corresponds to the coupled set of equations

x=h(x)+u

U=

(Z7a)

-T- 1U + T-l~(t)

where, as in (Z5a), ~(t) is a Gaussian white noise of zero mean and correlation function (~( t)e(t')) = ZD8(t t ' ) and T is the correlation time of the Ornstein­ Uhlenbeck noise, The associate Fokker-Planck equation is

a

atP(x, u, t)

a

= - ax ([h(x) + u)P(x, y, t)) 2

a T -1 P ( x, y, t ) + T -2 D au a 2 P ( x, + au

y, t ) ,

(27b)

For this case, as the difusion matrix is singular, we need to resort to a path integral representation in phase space, which (in prepoint discretization prescription) is

Non-Adiabatic Elimination of Variables in Stochastic Processes. . .

689

given by

P(Xj, Uj, tjlxo, UO, to) =

J

D[x]D[Px]

J

D[u]D[Pu]

+ ipx(;; - h(x) - u) -

7-

1

dt[ipu(-u + 7- 1 U)

2

Dp;].

(28)

As the dependence on Pu is Gaussian, the integration over this variable is immediate and afterwards we are also able to integrate over u. In order to do the last integra­ tion, we use known results for a charged oscillator in an external electric field. 5 ,13 If we made the margination procedure, which means integrating over the initial (stationary) distribution of u and over its final values, we obtain for the influence functional

D

rtf dt'lortf dt"p(t')e-

F[x, p] ~ exp ( - 2710

11

T -

I

t -t

"I

p(t")

)

(29)

which again has the same form as the proposed Ansatz (Eq. (22c)) with h == H = O. The result (29) is what one could obtain in a direct way by using Phytian's approach. 6b It is possible to integrate over p's, and to obtain the propagator in configuration space,14 but this point is not relevant here. We would like to remark that up to here we said nothing about time scales, all results are exact. However, we can consider a short correlation time (7 small) and expand in powers of 7. Up to zero order in 7, the integral in p in trivial and we recover the result of the propagator for the white noise problem in the prepoint prescription. A more complete discussion and analysis of this problem is done in Ref. 14. We turn now to the next example.

3.3. Nonlinear noise The possible origin of situations with nonlinear noise effects has been discussed in Ref. 15. The structure of the stochastic differential equations we will consider for this case is one where the noise enters quadratically (30a) where a is a parameter that fluctuates with Gaussian statistics and mean value

a

(30b) We assume now that TJ is an Ornstein-Uhlenbeck process, governed by an equation similar to the second one in Eq. (27a). Then the set (30a) could be rewritten as ;; = ~(x)

ry =

+ a 2 g(x) + 2ag(x)TJ(t) + g(x)TJ(t)2 -7- 1 TJ + 7-2~(t).

(30c)

I

690

H. S. Wio et al.

We adopt the following notation

& 6(x)+g(x,1])

n= -r­ 1] + r­ 1

where

2

+ a 2 g(x) + Dr- 1g(x) 1]) = 2ag(x)1](t) + g(X)1](t)2 - Dr- 1g(x).

6(x) = ~(x) g(x,

(3la)

e(t)

(3lb)

The mean value of g(x, 1]), when averaged over 1], is

For the set (3la) we can write the path integral representation of the propagator (again in the prepoint discretization) as

(32a) It is easy to perform the Gaussian integral over Pry' and to obtain

P (Xf, 1]f, tf Ixo, 1]0, to ) =

J

DxD [1 [px 1e

J dt[iPx(~-6(x))]

We could now use the Ansatz (22c). As in Ref. 15 we consider the weak noise approximation, which in terms of the simple case (lOa) is equivalent to a "weak­ interaction", and we can identify

ho,x(x) = 6(x)

j

(33a)

Also, for the averages in Eq. (23a) and (b), and as the simplest way of estimating these quantities in terms of known ones, we assume that

= ((g(x, 1]))) c::: (g(x, 1])) = 0 G(t, t') = ((g(x, 1](t))g(x, 1](t')))) c::: (g(x, 1](t))g(x, 1](t'))) = g(x(t))g(x(t'))(4a 2 r-1 De-lt-t'l/r + 2(r- 1 D)2e-2It-t'llr). hex)

(33b)

Non-Adiabatic Elimination of Variables in Stochastic Processes. . .

691

The final result for the marginal transition probability is

P (Xf, 1}f, tf Ixo, 1}o, to ) =

J [l

1

J dt[ip~(~-G(x»]

1) x 1) [px e

x e-t J dtJ dt'p~(t)c(t,t')Px(t')

.

(34)

To the lowest order in T we have that Te-Is-s'l/T :::::: 20(s - s'), and replacing this in (34) we recover the propagator associated with the Fokker-Planck equation found in Ref. 15. 4. Discussion

In this paper we have presented a non-adiabatic scheme of elimination of variables from multivariable Fokker-Planck equations or from a set of coupled Langevin equa­ tions inspired on Feynman's influence functional method, and have proposed Ansatz for such a functional. We have shown that the proposed Ansatz, in configuration or in phase space, can be justified, at least in those cases with weak coupling, via a perturbative expansion. Here, we would like to remark that this is true as far as the interaction Lagrangian is linear in the relevant variables and/or its velocities. For the more general case (at least in the configuration space representation) it is easy to see that the form of the Ansatz could be the same, but with the functions of the relevant variable or its velocity replacing them. Also, the Ansatz can be interpreted as the effect of colored but still Gaussian noise coming from the eliminated variables. However, if it is necessary to consider higher perturbational orders, in principle, we can generalize the Ansatz in order to include non-Gaussian effects. The examples we have examined in the last section show the great possibilities of this scheme. The first two have shown that the form suggested for the Ansatz arises naturally in cases where it is possible to obtain exact results, in configuration as well as in phase space. It is worth remarking that the case of the simple model is one where we do not have a clear separation of time scales. On the other hand the last example, the nonlinear noise case, shows that by using the Ansatz, together with some physical intuition about the form of the required correlation functions that appears in it, we can reach meaningful results. Another example worth considering, is the transition from Kramers description to the Smoluchowskii one in the high friction limit of Brownian motion. However such a case merits a special treatment and will be discussed elsewhere. 16 The emphasis of the presentation lies in the form of the solution, because at the very end this is the quantity of interest. In principle, it can be evaluated in some analytic approximation (saddle-point, etc.) or numerically. Nevertheless, there could be interest in knowing the form of the reduced equation. Even though this point has not been addressed here, what we expect is a kind of generalized Fokker­ Planck equation,17 including non-Markovian terms such as memory kernels. A deeper analysis of this problem is still lacking as it is a subject under investigation.

692

H. S, Wio et al.

Acknowledgments

One of the authors (H.S.W) greatly acknowledges fruitful discussions with M. San Miguel, E. Tirapegui, M. A. Rodriguez and L. Pesquera; and would like to thank for the kind hospitality extended to him during his stay at the Departamento de Ffsica, Universitat de les Illes Balears, Palma de Mallorca, and Departamento de Ffsica Modern a, Universidad de Cantabria, Santander, Spain. This work was partially performed with support from Fundacion Antorchas and CONICET, Argentina; and Direccion General de Investigaci6n Cientifica y Tecnica, Spain. Appendix A

In this appendix we present a short comment on some of the properties of the stochastic influence functional. (i) Indetermined situation such that only a probability distribution of the possi­ ble values that some parameter can adopt is known: Itlp, as well as the corresponding influence functional associated with each value: F p , then Feff

==

2: ItlpFp == (F) .

(A.1)

p

(U) Consider k (statistically independent) systems acting on the relevant one, and we know that each one influences the relevant system through Fk, then

(A.2)

In case of having Fk

:=

exp

{J dtf3k [x, ;;] , we obtain (A.3)

Other properties, as those indicated in Ref. 5b, as (iii), (iv) and (v), seems to have no equivalence in the present case. However, we still have a causality relation. Appendix B

In this appendix we want to present the exact results for the model described by Eq. (25a). The Lagrangian for this case has the form

which is quadratic in all the variables. Then, the path integral can be carried out exactly, for instance by calculating 13 P(Xf, Yf, tflxo, Yo, to)

= (21f)-ldet(82 Smp [x, y]/8(Xfy/)8(xoYO)))1/2 x e-s=p[x, y]

(B.2)

Non-Adiabatic Elimination of Variables ir.

::::~chastic

Processes. • .

693

where Smp[X, y] is the action evaluated along the most probable path, that means along the path that fulfills the Euler-Lagrange equations associated with the La­ grangian (A.I). We also need to evaluate the Van Vleck determinant in the prefac­ tor. Such calculations lead us to the following result P(Xf, Yf, tflxo, Yo, to) = (21TDT)-le-(2DT)-1I R r

where T = tf

to, Rf

= (xf, Yf)' Ro = (,p,

Ro I2

(B.3)

JL), and

+ Yo sin(bT) JL = -Xo sin(bT) + Yo cos(bT) . ,p

Xo cos(bT)

Now doing the margination procedure, where we assume for the initial distribution that P(Yo) = o(Yo Yo), we get

Now, ifin order to simplify the expressions, we adopt Yo = 0, to = 0 and tf = t, it is easy to prove that the above indicated inclusive of marginal conditional probability is a solution of the following equation

:t-Rncl(Xf, tflxo, to)

= bxosin(bT) !-Rncl(Xf, tflxo, to) 1

Ej2

+ 2D aX2 -Rncl(X f'

tflxo, to) .

(B.5)

The form of the drift term, with its dependence on the initial condition as well as on time, indicates the non-Markovian character of the equation. References 1. H. Haken, Rev. Mod. Phys. 41, 67 (1975); H. Haken, Synergetics: An Introduction

(Springer-Verlag, Berlin, 1977). 2. Among others: C. W. Gardiner, Phys. Rev. A29, 2814 and 2823 (1984); W. Theiss and U. M. Titulauer, Physica 130A, 123 and 143 (1985); N. G. Van Kampen and I. Oppenheim, Physica 138A, 231 (1986); D. Wycoff and N. L. Balaz, Physica 146A, 175 (1987). 3. N. G. van Kampen, Phys. Rep. 124, 69 (1985). In this review there is an extensive list of references on adiabatic elimination procedures. 4. L. E. Reichel and W. C. Schieve, Instabilities, Bifurcations and Fluctuations in Chemical Systems (Univ. Texas Austin, 1982); K. Kitahara, H. Metiu and J. Ross, J. Chem. Phys. 64, 292 (1976) and 65, 393 (1976); H. A. Weidenmuller, Prog. Partie. Nucl. Phys. 3, 49 (1980). 5. (a) R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw Hill, New York, 1965); (b) R. P. Feynman and F. 1. Vernon, Ann. Phys. (N. Y.) 24, 118 (1963). 6. (a) L. Pesquera, M. Rodriguez and E. Santos, Phys. Lett. 94A, 287 (1983). (b) R. Phytian, J. Phys. A10, 777 (1977).

694

H. S. Wio et al.

7. H. S. Wio, "An influence functional approach for the elimination of variables in stochas­ tic processes", in Noise and Nonlinear Phenomena in Nuclear Systems, eds. J. L. Munoz Cobo and F. Defillipo (Plenum, New York, 1989); H. S. Wio, C. B. Briozzo and C. E. Budde; Elimination of variables in stochastic processes: a path integral point of view; in Lectures on Path Integration: Trieste 1991; eds. H. A. Cerdeira et al. (World Scientific, Singapore, 1992). 8. F. Langouche, D. Roekaerts and E. Tirapegui, Nuovo Cim. 53B, 135 (1979); R. Graham, Z. Phys. B26, 281 (1977). 9. F. Langouche, D. Roekaerts and E. Tirapegui, Functional Integration and Semiclassical Expansions (D. Reidel Pub. Co., Dordrecht, 1982). 10. It is worth remarking that Feynman's influence functional is, in fact, a double path integral. 11. D. M. Brink, Prog. Partie. Nucl. Phys. 4, 323 (1980); K. Mohring and U. Smilansky, Nucl. Phys. A338, 227 (1980); X. Wu, Z. Li, J. A. Maruhn, W. Greiner and Y. Zhuo, J. Phys. G14, 1049 (1988). 12. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North Holland, Amsterdam, 1981); C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1983). 13. S. Khandekar and S. Lawande, Phys. Rep. 137, 115 (1986); L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981). 14. P. Colet, H. S. Wio and M. San Miguel, Phys. Rev. A39, 6094 (1989); H. S. Wio, P. Colet, M. San Miguel, L. Pesquera and M. A. Rodriguez, Phys. Rev. A40, 7312 (1989). 15. M. San Miguel and J. M. Sancho, Z. Phys. B43, 361 (1981); F. Sagues, M. San Miguel and J. M. Sancho, Z. Phys. B55, 269 (1984); F. Sagues and M. San Miguel, Phys. Rev. A32, 1843 (1985); W. Horsthemke and R. Lefever; Noise Induced Transitions (Springer-Verlag, Berlin, 1984) Ch. 8.7. 16. H. S. Wio, to be submitted. 17. J. Nordholm and R. Zwanzig, J. Stat. Phys. 13,347 (1975); H. Risken, The Fokker­ Planck Equation (Springer-Verlag, Berlin, 1983).