Mesoscopic stochastic descriptions of many particle systems have frequently been used in the context of mean field nonlinear Fokker-Planck equations (Desai and Zwanzig 1978; Kuramoto 1984) and nonlinear Fokker-Planck equations related to nonextensive entropies (Plastino and Plastino 1995). We discuss a recent attempt to obtain a unified description for stochastic relaxation processes that are characterized by mean field interactions, on the one hand, and nonextensivity, on the other hand (Frank 2001b).
First, we consider the stationary case. To this end, we introduce the concept of inverse distortion functions (Frank, Daffertshofer 1999). For mean field models we obtain from inverse distortion functions implicit descriptions of stationary distributions that involve transcendent equations and can be used to address the phenomenon of multistability. For nonextensive systems we obtain cut-off and power-law distributions.
Second, we examine the transient case. We briefly discuss exact time-dependent solutions of systems related to the Sharma-Mittal entropy (Frank, Daffertshofer 2000). Then, H-theorems are developed on the basis of (i) inverse distortion functions and (ii) free energy measures (Shiino 1987,2001; Bonilla et al. 1998; Frank 2001a,b,2002; Kaniadakis 2001).
Third and finally, we study the stability of stationary distributions for a stochastic mean field model that is in line with the field theoretical description of neural activity proposed by Haken (1996) and Jirsa and Haken (1996) and can describe neural activity during rhythmic finger movements (Frank et al. 2000). We focus on two approaches: the transcendent equation approach and Lyapunov's direct method (Frank et al. 2001).
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