stochastic differential equation is derived by means of the Kramers-Moyal method, and will be ... equations for the probability density P (x, t) were of the general.
Physica
85A (1976) 363-373
TIME-LOCAL
0
North-Holland
GAUSSIAN
Publishing
Co.
PROCESSES,
AND NONEQUILIBRIUM
PATH INTEGRALS
NONLINEAR
DIFFUSION
H. DEKKER Physics
Laboratory
TNO,
Dm Haag, 7he Netherlands
Received 14 April 1976
In this paper we discuss the concept of time-local gaussian processes. These are processes for which the state variable at time t + T is gaussian distributed around its most probable value at that time, for a specified realization a small time interval t earlier. On one hand it will be shown that these processes are related to a very simple path sum. On the other hand the associated stochastic differential equation is derived by means of the Kramers-Moyal method, and will be seen to be the most general nonlinear Fokker-Planck equation. The significance of the present formulation for nonequilibrium processes and the comprehension of critical phenomena will be evaluated.
1. Introduction It has been known already for some time that striking analogies exist between many different physical systems in thermal quilibrium near critical points. One only needs to mention the unifying theory of Ginzburg and Landau’) for second order phase transitions and the more recent approaches to critical phenomena by means of the renormalization group** “). Only during the last decade the notion has grown that such a unification also applies to quite varying systems which are L aser threshold’-lo), far from thermal equilibrium4-6). BCnard’s convection instability in fluid dynamic& 11), c h emical instabilities”), and the GinzburgLandau theory for continuous nonequilibrium systemsg* lo* 12-16) are by now well-established examples. The most profound analogies far from equilibrium have been found in the realm of the Fokker-Planck description (e.g.g* lo* “-‘l)). The phenomena at critical points are essentially connected with nonlinear equations”) and, therefore, the Fokker-Planck equations for the probability density P (x, t) were of the general type : ;
p
(4 t> = -
$
[Cl(X)
p(4 01 + 3 5 363
[c*(x) P (x,
t)] ,
364
H. DEKKER
where the drift coefficient cl(x) and the diffusion coefficient cZ(.x) are arbitrary functions of the systems coordinate X. One can arrive at such a Fokker-Planck equation in two alternative ways. First, one may expand the Chapman-Kolmogorov-Smoluchovski or master equation, either classical or quantum mechanical (see e.g. 6, 22-25)), in differential form and crudely break off after the second derivative. This diffusion upproxi~afion is said to describe continuous Markov processes. The other method starts from the generalized Langevin equation for the dynamics of the fluctuating coordinate itself (E.~.~,~O,~~)). The quantum mechanical version of the Langevin equation is also known as the noise operatol equationz6). Using standard techniques (~.g.~~)) one may convert these Langevin equations into their stochastically equivalent Fokker-Planck equations. The first method has received considerable criticism in the sense that the general nonlinear Fokker-Planck equation would not be a systematic approximation to the fundamental master equa&ion22* 2h). In the realm of Van Kampen’s systematic equations occur, that is system size expansion 22. 26) only linear Fokker-Planck those where C,(X) is linear in .Yand c,(x) is state independent. However, Van Kampen’s method, implicitly containing a “central limit theorem” argument, breaks down essentially at critical point where the size of stochastically independe:?t fluctuating domains becomes comparable with the system size and, hence. their number is relatively small’“. 27). Moreover, a number of nonequilibrium critical phenomena, such as occur for example in nonlinear opticsz8), fundamentally depend on finite geometric?, 29) while Van Kampen’s linear Fokker-Planck equation only occurs in the so-called thermodynamic limit of infinite system size. The Langevin method has been criticized since it strongly depends on a proper choice of coordinates and fluctuating forces whether the Langevin description fits into the framework of the master equation even for linear systemsz6* 30-32). Another drawback of the method is that, due to the Markov properties of the fluctuating forces t(t), the actual process x(t) is nowhere differentiable and the Langevin equation can in fact be interpreted only in the sense of generalized of stochastic processes 33. 34). These difficulties have even lead to the development the mathematically more refined Ito equations35). Moreover, for multivariate processes the Langevin description leaves a degree of freedom in the choice of the fluctuating forces”*) which cannot be understood very well from the physical point of view. Even for onedimensional problems with nonconstant diffusion function c~(_Y)the state of the noise sources becomes dependent on that of the system, namely (cj(s) :(I)) = t (ac, (X)/&Y). Though formally correct, this property seems to be physically somewhat unreaIistic36* 37). In the present paper we consider the physically plausible concept of time-local Gaussian processes. To obtain a better grasp of its value we first briefly discuss in section 2 a linear Gaussian diffusion process. Then in section 3 we define the nonlinear, but time-local gaussian process. Linear time-local Gaussian processes remain gaussian also for fii;lite time periods. It will be shown on one hand that the
GAUSSIAN
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concept leads to a quite simple sum over pathsz8* 34*38-40). In section 4 on the other hand, starting from the fundamental master equation, we shall demonstrate that it leads exactZy to the most general nonlinear Fokker-Planck equation (1). In section 5 we briefly discuss why the concept of time-local gaussian processes may elucidate the fact that nonlinear Fokker-Planck equations have proven to describe critical phenomena in such an accurate way. Finally, in the Appendix we touch upon the conversion of the discrete path sum presented in section 3 into a path integral, a subject which has not been settled yet2*, 34,41-46).
2. Linear gaussian processes Let us briefly consider a fluctuating process x(t) with a linear macroscopic equation 1 = cl(x) = clx and constant diffusion coefficient cl. Such a process is usually called a Uhlenbeck-Ornstein process47), although it was first studied by Smoluchovski48) for the case of a brownian particle under influence of outside forces, for example gravitation. The free brownian diffusion with zero macroscopic drift, that is i = 0, has been studied by Einstein4g), but is usually called a Wiener process3s) in mathematical physics. The Fokker-Planck equation for the linear Uhlenbeck-Ornstein process reads:
-%(x,t)= C7t Its fundamental
Greens
function
1V (s. t/x0, 0) = [2n02]-’ where we have carefully macroscopic dy/dt
(2)
-r,~rP(x,t)+-~~,~P(x,t).
solution exp [-(x
be20,
24.43,47,50
1:
- y)z/20z] between
(3)
the actual
realization equation:
N and the
= c,y
and thus here yields y (t/O) = x0 exp (c, t). The variance of motion of the second moment: da’ldt
to
value y(t) which follows from the macroscopic = cl(y)
distinguished
is well-known
(4) follows from the equation
= c2 + 2c,02
with O’(O) = 0. Hence, for the present case one obtains explicitly: = ( -c2/2c,) [l - exp (2c,t)]. The distribution (3) is gaussian due, first the diffusive character of (2), which is intimately connected with the character of the fluctuating forces which give the macroscopic process its (3) remains ical character (see e.g. 23*28, 47--51)). And the distribution
(5) o2 (t/O) of all, to gaussian stochastgaussian
366
H. DEKKER
for any time period t as a consequence of the linearity of the processsl). Any nonlinearity in the macroscopic coordinates destroyes the gaussian character 28, “2)). (e.g. Let us now specify the Greens function (3) for an infinitesemal time interval Z: W (x, z/x0, 0) = [2xc2 (t/O)]-’ From the general y (t/O) =
.Yg
u2 (z/O) = Inserting
solutions +
zc2
(6)
of (4) and (5) we infer that:
(XCJ+ 0(x2),
zc1
+
exp { - [X - y (t/0)12/2u2 (r/O)}.
(7)
O(t’).
(8)
(7) and (8) into (6) we may write:
W(x, t/x,,
0) = [~TcTc,]-~ exp { - [s - so - tcl (x~)]~/~w,).
(9)
In the limit that z tends to zero we thus have a gaussian distribution about the socalled prepoint x0 1401 with variance zc2, that is the only appreciable transitions away from the prepoint are of the order z’. Of course, the limit for z going to zero is a mere mathematical concept. It should be kept in mind that from the physical point of view z is some intermediate time. On one hand it must be much smaller, of course, than the characteristic relaxation times of the macroscopic process, but on the other hand it still should be so large that the macroscopic coordinate receives many independent pushes from the microscopic fluctuating forces (see e.g. 6, I93 49*53)). In other words, the physical z should be large compared with the correlation time of the noise sources. If we specify a sequence of gaussian factors like (9) we obtain the multigate transition probability to pass through the specified chain x,, at to, x1 at t, = to + z, ending at s,, at t,. Integrating over all possible intermediate {xi} one arrives . .. at the sum over paths between W({X,,>) = .i 7
x0 and X, (see also40-43,
[2rctc2]- ’ dsi exp [ -$
5’)):
tG (xi; xi_ i)]
where : G(si;xi_l)
=
2
$-
c,(.q--1) 2
.
(11)
1
It should be remembered that we have derived (10, 11) on the base of the Greens function (3) for a linear process. As mentioned before, nonlinear processes do not have a gaussian distribution like (3) for finite time intervals (see e.g.“)).
GAUSSIAN
PROCESSES,
3. Nonlinear time-local
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367
gaussian processes
In order to generalize the above considerations to general nonlinear processes where bo;h the drift and the diffusion function are state dependent in an arbitrary way, it will be profitable to realize that although the distribution will not be a gaussian on a global time scale, it may be reasonable to assume that it still remains gaussian indeed over very small time intervals Z. Firstly, as long as z is large compared to the fluctuation correlation time the coordinate receives many independent pushes during this time interval, which is a prerequisite for a gaussian distribution. Secondly, we have seen that as t becomes vanishing small with respect to the process relaxation time the variations in X become very small too, namely of order ,*, and the equation of motion for an infinitesimal variation, say F = 6X, is always linear: & = [&, (X)/ax] F. Thus it seems rather reasonable to assume that many physical processes are gaussian distributed locally in time, although this property is lost on a global time scale due to nonlinearities. Let us therefore generalize (9) to the class of these time-local gaussian processes as: M/(x,, ti/xi-
1, tr_1) = [~XZC, (X-r)]-‘exp
[-ZG(Xi;Xi_l)]
(12)
with : G (xi; xi-I)
1
= 2c2
xi -
xi-
2 1
- cl(xi-l)
(13) 1
z
Cxi--l)
where -r = ti - ti_-l, and where now both cl(X) and cl?(x) are arbitrary functions of X. It needs emphasis that the otherwise logical choice of both functions at the prepoint xi-l for each infinitesimal transition will be shown to be of crucial importance (see also40)). The definitions (12) and construct the path sum analogous to (10):
W((X~}) = J ndXiW(Xi,
ti/Xi-l,
which now is defined for general
4. The Fokker-Planck
(13) may again
ti-1)
nonlinear
be used to
(14) diffusion
processes.
equation
In this section it will be shown that the path sum (12-14) leads in an exact manner to the general nonlinear Fokker-Planck equation (1). The nonlinear Markov process described by (12-14) obeys the master equation : P(x, t + t) = jdx,W(x,
t + z/x0, t)P(x,,
t)
(15)
H. DEKKER
368
where we have denoted the prepoint by x0 = xi_1 and the postpoint by s = si for convenience. The transition probability for the time-local gaussian process can be gleaned from (12). We now invoke the we!l-known Kramers-Moyal expansion of the master equation, which reads in generalz3r 4”, 54) :
f)=i;-L n=1
‘I[k, (x0,
n!
where it has been assumed
t) P (so, t)]
that the limits of the transition
moments:
k, (x, , t) = lim k, (x0, t, z) 710
(17)
exist, and where: k, (x0, t, t) = f
dx (X - ,vo)” W(x, t + z/x-o, t).
It is noted that the k,, will be time independent if the process is homogeneous in the sense that W(x, t + z/,yo, t) does not explicitly depend on t. Inserting now the square in the exponential and setting s - so (12, 13) into (18), evaluating = zfz, with dx = rf dz, one finds:
k,, = ?-l
[2nc,lAs ’ exp (-2)
rexp(-$
+ :il;)z”dz
(19)
-m where in view of the definition of all functions allowed to drop the argument _yo. The integral to give: [n/21
k, = T”-~c;H!
m=~
m
1
1 (72
-
at the specified prepoint we were (19) can be calculated (see e.g.““))
2m)! m!
where the symbol [n/2] stands for the integral part of the real number now z tend to zero it is easily seen that all k, vanish except: k,(xo)
=
c,(xo);
k&o)
=
4x0).
(20) n/2. Letting
(21)
Substituting this result for the moments into the Kramers-Moyal equation (16), this is immediately found to reduce precisely to the general nonlinear FokkerPlanck equation (1).
GAUSSIAN
PROCESSES,
PATH INTEGRALS
AND DIFFUSION
weighted
over all possible
369
5. Some concluding remarks The interesting
idea of properly
summing
alternative
routes which a system can follow in its evolution has great intuitive and suggestive power. Wiener3’) was the first to study the method in his investigations on the brownian motion. Feynman39r 56, 57) op,-ned up new vista’s by demonstrating the value of the concept in quantum m:chanics. His work lead to the development of an extensive mathematical apparatus for functional integration56-61). Onsager and Machlup’l) should be mentioned for their pioneering work in applying the notion of summing over paths in the theory of irreversible, nonequilibrium thermodynamics62). Unfortunately, their approach was restricted to linear processes as was the work of Falkoff63) on the Uhlenbeck-Ornstein process47). The restriction to linear drift was lifted only less than a decade ago by Wiege141), but the diffusion function still remained state independent46). In the introduction of the present paper the excellent description of nonlinear nonequilibrium phenomena by means of the nonlinear Fokker-Planck equation has already be-n mentioned. We signalized there the discussion about the fundamental value of the nonlinear Fokker-Planck equation. To get a better comprehension of the principles underlying the general Fokker-Planck equation, we have introduced the physically plausible concept of time-local gaussian processes in sections 2 and 3. Linear processes preserve their gaussian distribution on a global time scale, but this property is destroyed by nonlinearities. We have shown in section 4 that indeed the concept of time-local gaussian processes underlies the general nonlinear Fokker-Planck equation. In the vicinity of critical points we quite generally observe the phenomenon of slowing down (see e.g.6s ‘3 17*18*28)), such that in these regions it should be quite easy to define a small time scale z being long compared with the correlation time of the fluctuating forces, but much shorter than the process relaxation time. Hence, the time-local gaussian concept should apply very well. In section 4 we have in fact demonstrated the relation bztwzen the general nonlinear Fokker-Planck equation (1) and the path sum (12-14). This formulation may be considered as an extension of the Onsager-Machlup theory to nonlinear processes, which may open up new ways to a general formulation of nonequilibrium thermodynamics. It should be remarked that the present generalized formulation violates the fifth postulate of the original approach as presented by Tisza and Manning (“‘); see also2*)), for the measure in function space g{xi} = Hi [~XZC, x (xi- ,)I-” dxi may now depend on the state of affairs through cz(x). The discrete path sum formulation (12-14) is properly defined. A concluding remark then concerns an effort to formulate the path sum (12-14) as a convent ional path integral for nonlinear processes (see e.g.41 -““)). Replacing that integral correctly through first order in z by its equivalent sum expression would lead to a stochastic differential equation unlike (1) and, hence, to different physics, even for
H. DEKKER
370
a constant diffusion coefficient. We have devoted and will return to it in extenso in another paper.
our Appendix
to this subject
Appendix. On the relation between path sums and path integrals for nonlinear processes The formulation of the problem might be as follows: we wish to rewrite the well-defined sum over i in the exponential in our path sum (12-14) (see also lo), as a conventional integral over t without further specifications, subject to the requirement that both formulations lead to the same associated differential equation (1). We shall confine ourselves at present to processes with constant diffusion coefficient. Since in (15) on the left hand side P (x, t + t) = P (s. t) + z aP/dt + O(z’) we conclude that we must evaluate all quantities occurring at the right hand side about the postpoint at least through order Z. This idea leads to the so-called Feynman method39* 40, 56) f or deriving the associated differential equation from (15). We have already noticed in section 3 that when z becomes vanishing small, si - si_r = Ci becomes of order r3. Evaluting the square in the exponential in W(.ui_ 1 + Ci, ti_ , + t/xi-I, ti_ 1) we find three factors. The first is the dominating one: exp [-
