Quasilinear Fokker-Planck equation

We shall call this a quasilinear Fokker-Planck equation, to indicate that it has the form (1.1) with constant B but nonlinear It is clear that this equation can only be correct if F(X) varies so slowly that it is practically constant over a distance in which the velocity is damped. On the other hand, the Rayleigh equation (4.6) involves only the velocity and cannot accommodate a spatial inhomogeneity. It is therefore necessary, if F does not vary sufficiently slowly for (7.1) to hold, to describe the particle by the joint probability distribution P(X, V, t). We construct the bivariate Fokker-Planck equation for it. 

It is a bivariate quasilinear Fokker-Planck equation of the form (6.1) with semi-definite fJ. 

We call this equation quasilinear to express the fact that the coefficient of L(t) is still a constant. Although its solution cannot be given explicitly it can still be argued that it is equivalent to the quasilinear Fokker-Planck equation... 

In the case of the quasilinear Fokker-Planck equation (2.4), the free energy U defined in terms of the stationary solution by (2.6) is identical with the potential in the deterministic equation (5.2). That identity is often taken for granted when time-dependent solutions have to be constructed for systems of which only the equilibrium distribution is known. We shall now show, however, that it holds only for systems of diffusion type whose Fokker-Planck equation is quasilinear, i.e., of the form (2.4). 

So far we studied instabilities in discrete one-step processes. Now consider diffusion processes as in XI. For simplicity we restrict ourselves to diffusion in an external potential U(x), as described by the quasilinear Fokker-Planck equation (XI.2.4) ... 

For convenience we restrict the discussion to the quasilinear Fokker-Planck equation (1.8). The corresponding results for the one-step process will be left for Exercises. Our U(x) has the shape of fig. 36b the deterministic equation has the same form as a10 in fig. 34. 

The only way to obtain a well-defined and physically meaningful approximation is by performing again an expansion in powers of a physical parameter. If the lowest order is to be deterministic the parameter has to be such that for small values of it the distribution reduces to a narrow peak. Clearly the parameter 6 = kT is suitable, because the low temperatures have small fluctuations. We shall show that the same method used in X for obtaining the -expansion can be adapted to obtain an expansion of the Fokker-Planck equation in powers of 01/2. We first demonstrate the method for the one-variable quasilinear equation (2.4). 

Exercise. Any nonlinear Fokker-Planck equation (1.5) can be transformed into a quasilinear one by a suitable transformation of x. 

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