Exact temporal evolution for some nonlinear diffusion processes

Exact solutions to Fokker-Planck equations with nonlinear drift are considered. Applications of these exact solutions for concrete models are studied. We arrive at the conclusion that for certain drifts we obtain divergent moments (and infinite relaxation time) if the diffusion process can be extended without any obstacle to the whole space. But if we introduce a potential barrier that limits the diffusion process, moments converge with a finite relaxation time.


I. INTRODUCTION
A time-dependent Fokker-Planck equation (FPE) describes the dynamical evolution of the diffusion processes.Nevertheless, when the dynamics of the process is nonlinear it is very difficult to obtain exact or even approximate solutions of such FPE's.Since at the present time nonlinear processes are ofhi8hest interest (instabilities, phase transitions, etc.)• many people have tried to find exactly soluble nonlinear models.2-7 The most common technique used to solve exactly a FPE consists in separating the temporal from the spatial dependence; this latter one is solved by means of an eigenfunction expansion in the same way as occurs with the SchrOdinger equation.[2][3][4] Another more direct although more skillful technique is initiated in Ref. 6 and continued in Ref. 8. It consists in separating the part which is most related to the potential of the process (which causes the nonlinearity) from the probability density P (q,t ); then, by means of convenient assumptions, the remaining part of P (q,t ) is solved separately assuming that it is Gaussian.Concretely in Ref. 8 we have found that the N-dimensional FPE P(q,t) = -al '[p'(q).P(q,t)) +~al' al'p(q,t), (1.1) when al'=a/aql' (sum over repeated Greek indices is assumed) has an exact solution, with the usual initial condition if the drift/I'(q) = /1' (q.,q2, ... ,qN) has the form for whatever value of the dimension N of the phase space, and if N is odd.In these cases the normalized solution of the FPE is given by (1.5) with t,6 (q) given by (1.10) or (1.11) and b =(4/-N}a. (1.12) In this article we intend to study some of the applications of these solutions for concrete models.

II. A FIRST MODEL OF NONLINEAR DIFFUSION
By means of an adequate selection of the constants ak andPk that appear in (1.4) we can write t,6 (q) = 1 kU.{e(a I 2J<dD -21)v'20qk)} I, For t~o the functions D_I,(..j2Qqd do not have real zeros and instead of (2.1) we can write The characteristic function e (.u I'.••' I'-N ) associated to a density of probability P(q,t) is given byl (2.4) Substituting in (2.4) the probability density given by Eq. (1.5) with ¢ (q) given by (2.3) we have ,(..j2Qqk) Once we have evaluated the characteristic function, moments follow easilyl: (2.9) and (2.11)In this model the drift may be written in the form (2.12) The first moment (2.10) as a function of the drift is (
The case I = 0 corresponds to linear drift and, therefore, presents no difficulty.
We come to the conclusion that both models presented in this section could not be valid for the study of the temporal evolution of physical systems towards equilibrium.In the following section we find a mechanism that yields exactly soluble models that relax towards equilibrium with a finite relaxation time.

III. ONE-DIMENSIONAL MODEL WITH POTENTIAL BARRIER
For a one-dimensional system the potential V(x) of the drift (1.3) is where f/J (x) is given by Eq. (1.4).By means of an adequate choice of the constants a k andf3k' we can write where U(/I!lax 2 ) is a function of Kummer.9Let us suppose now, that, for a certain value of XI <x o , there exists a potential barrier, that is, which is equivalent to the following expression for f/J (x) f/J (x) = {U(/I~lax2), X;;;'XI' This is possible since f/J (x) = 0 is also a solution of the differential equation that satisfies the function (3.2) (see Ref. 8).
We will finish this section studying the case when the potential barrier is very far away from the origin (that is, our initial state), i.e., when axi>1. (3.12) In such a case, as in Ref. 10, U(alclz)-z-a for Z-oo (Rea>O), (3.13) the functions rfJ' n m )(/,xI;7J(t)), defined by (3.8), may be written Taking into consideration 9 When t> 1120, we have 7J(t )= 1.With the approximation (3.13) we have ([x(t) (3.17) Therefore, even when the potential barrier is very far away from the initial state, the evolution of the system depends on the position x I of the barrier.
If we compare Eq. (3.16) with Eq. (3.10), and Eq.(3.17) with Eq. (3.9), we observe that the asymptotic temporal evolution of the model is similar to the evolution of the general case.
For x <XI the potential (3.3) is a hard-core potential.
This implies that the barrier is a reflecting barrier.Thus, the probability current J (x,t ) must be zero for x<x I' In our case J (x,t ) is given by (3.18) where and 9 Thus J (x,t ) will be zero at the barrier if ¢ '(x) is a continuous function at x = x I' This implies that the potential barrier must be located at the zeroes of the Kummer function.
If, instead of (3.4), we write ¢ (x) = {U(I 1!lax 2 ), x;>x l , a, x<x l , where a<1 (our potential is not completely hard core), we have P(x,t /O)~O for x<x I (3.20) [see Eq. (3.5)] and the barrier may be located anywhere.In Fig. 2 we have a representation of the potential (3.3) in the case where I = -0.5.

IV. STATIONARY DISTRIBUTIONS V(x)
As is well known a one-dimensional FPE  (4.5) where and (provided that the proper choice of the constants a and p extends this normalization to the models (4.6b) and (4.6c)].
Let us study the stability of these stationary distributions.Following the criterion given in Ref. 4 we can affirm that the stochastic process represented by Eq. ( 4.1) has a stable stationary solution, and all moments (xm) up to the mth order exist if the following inequality is satisfied: > .
x_", (m + l)lnx (4.8) In Thus, we can affirm that the models presented in Sees.II and III are stable for any value of the parameter I.

V. CONCLUSIONS
Relating the results of Sec.II with those of Sec.III, we observe that the nonlinear diffusion process, represented in general form by the drift (1.3), yields divergent momenta (and infinite relaxation times) if the diffusion process can be extended to the whole physical space.Nevertheless when, due to the introduction of a potential barrier, the diffusion process takes place in a limited part of space, the moments converge with finite relaxation time given by 1"nonlinear = 1/21a.Comparing this relaxation time with the one that corre-526 J. Math.Phys., Vol. 26, No.3, March 1985 sponds to a linear drift,f Il(q) = aqll, that is, 1"linear = 11a, we observe that this process of nonlinear diffusion relaxes quicker than the linear diffusion if

I>! .
Let us remark also that the nondivergent model studied in this paper can reach large parts of space since the asymptotic evolution of the process is the same no matter how far away the potential barrier is from the initial state.
The general model represented by Eq. (1.4) is unstable since its stationary distribution Pst (x) diverges.The models studied in Sees.II and III are stable.