Nonlinear Fokker-Planck equation exhibiting bifurcation phenomena and generalized thermostatistics

摘要

A nonlinear Fokker-Planck equation exhibiting bifurcation phenomena is proposed within the framework of generalized thermostatistics. The nonlinearity responsible for the occurrence of bifurcation of solutions is assumed to be of the form appearing in the standard mean field model. A Liapunov function is defined that takes the form of free energy involving generalized entropies of Tsallis and an H-theorem is proved to show that the free energy, which is bounded below, continues to decrease until the system approaches one of the equilibrium distributions. The H-theorem ensures, instead of uniqueness of the equilibrium distribution, global stability of the system in that either one of multisolutions must be approached for large times. Local stability analysis is conducted and the second-order variation of the Liapunov function is computed to find its relevant part whose sign governs stability of the equilibrium distribution of the system. The case with a bistable potential is investigated, as an example of confirming the theory, to give the bifurcation diagram displaying the order parameter as a function of the coefficient of the nonlinear diffusion term. (C) 2002 American Institute of Physics. [References: 51]