Nonlinear Fokker-Planck equations associated with generalized entropies: Dynamical characterization and stability analyses

Dynamical characterization and behavior of two types of nonlinear Fokker-Planck equations (NFPEs) are studied within the framework of generalized thermostatistics. On the basis of generalized entropies NFPEs are shown to be constructed so that H-theorems hold with Lyapunov functionals that are given by free energy functionals associated with the generalized entropies. In the case of the ordinary type of NFPEs such H-theorems ensure convergence of solutions to their uniquely determined equilibrium solutions. In the case of mean-field type NFPEs (DNFPEs) that may exhibit bifurcation phenomena the H-theorems are shown to still hold to ensure global stability of solutions. Systematic description is given of local stability analysis based on the second-order variations of the free energy functionals.