Stability analysis of stationary states of mean field models described by Fokker–Planck equations

Abstract

So far, the stability of stationary solutions of mean field models described by Fokker–Planck equations has only been determined analytically in some special cases. Following two earlier studies [J. Korean Phys. Soc. 40 (2002) 1037; Prog. Theoret. Phys. Suppl. 150 (2003) 48], we discuss a stability analysis for a large class of mean field models based on Fokker–Planck equations. To this end, we use linear stability analysis in addition to the well-known approaches by means of transcendent equations and Lyapunov’s direct method. We demonstrate that all three methods yield consistent results for systems that exhibit free energy functionals (e.g., equilibrium systems). We show that the simple transcendent equation analysis fails for systems that do not exhibit free energy functionals (e.g., nonequilibrium systems) and show how to solve this problem by means of a more sophisticated transcendent equation analysis. Furthermore, we propose a norm for the perturbations of stationary states and illustrate some of our results by a model that exhibits a reentrant noise-induced phase transition.