The nonlinear Fokker-Planck equation with nongradient drift forces and an anisotropic potential.

Abstract

Studies regarding physical phenomena described by nonlinear Fokker-Planck equations usually consider the case where the drift forces acting on the physical system under investigation are derived from the gradient of a potential function. In the present manuscript, we investigate nonlinear Fokker-Planck equations, where the drift field has a component that is derived from the gradient of an asymmetric potential and another that corresponds to a nongradient force term. We consider the specific case of a two-dimensional, nonlinear Fokker-Planck equation where the drift field is obtained from an anisotropic, harmonic potential, besides the nongradient term. We analyze the conditions under which this evolution equation admits stationary solutions that are q-exponentials. We prove that this equation admits q-Gaussian, time-dependent solutions that evolve to stationary forms and discuss some of their important properties. Interpreting the evolving probability densities as describing an ensemble of particles, we derived and numerically studied the concomitant trajectories. The theoretical framework discussed in the present contribution enlarges substantially the range of possible applications of the Sq-based, nonlinear Fokker-Planck formalism to the study of problems in diverse types of complex systems in physics, biology, economics, artificial neural networks, and other areas. Importantly, it allows us to approach the problem of modeling systems where the interaction among elements are not symmetrical, as is the case, for example, in neural networks of the brain.