Fokker-Planck equation nonlinear problems

Suppose one is faced with a one-step problem in which the coefficients rn and g are nonlinear but can be represented by smooth functions r(n), g(n). Smooth means not only that r(n) and g(n) should be continuous and a sufficient number of times differentiable, but also that they vary little between n and n+ 1. Suppose furthermore that one is interested in solutions pn(t) that can similarly be represented by a smooth function P(n, t). It is then reasonable to approximate the problem by means of a description in which n is treated as a continuous variable. Moreover, since the individual steps of n are small compared to the other lengths that occur, one expects that the master equation can be approximated by a Fokker-Planck equation. The general scheme of section 2 provides the two coefficients, but we shall here use an alternative derivation, particularly suited to one-step processes. 

Chapter II addresses another fundamental problem under what physical conditions can the Fokker-Planck equation provide a reliable picture of fluctuation-dissipation processes Aware as they are of the technical and conceptual difidculties involved in nonlinear statistics, the authors share Zwanzig s optimistic view that use of the Fokker-Planck equation is practicable and advantageous. Chapter II contains a brief description of rules to construct, via a suitable procedure, equations of Fokker-Planck type for the slow variables of the system under study. This theoretical method eliminates explicit analytic dependence on fast variables and thereby produces a significant simplification of the problem under discussion. 

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