Curl Forces and the Nonlinear Fokker-Planck Equation

TL;DR

This paper explores nonlinear Fokker-Planck equations with curl forces, establishing conditions for stationary solutions as q-exponentials, and presents a two-dimensional model with analytical solutions relevant to complex systems like superconductors.

Contribution

It provides new conditions for stationary solutions in nonlinear Fokker-Planck equations with curl forces and analyzes a specific model with analytical q-Gaussian solutions.

Findings

Stationary solutions are q-exponentials under certain conditions.

The model describes particles with short-range interactions in a rotating medium.

The equations satisfy an H-theorem involving S_q entropy.

Abstract

Nonlinear Fokker-Planck equations endowed with curl drift forces are investigated. The conditions under which these evolution equations admit stationary solutions, which are $q$-exponentials of an appropriate potential function, are determined. It is proved that when these stationary solutions exist, the nonlinear Fokker-Planck equations satisfy an $H$-theorem in terms of a free-energy like quantity involving the $S_q$ entropy. A particular two dimensional model admitting analytical, time-dependent, $q$-Gaussian solutions is discussed in detail. This model describes a system of particles with short-range interactions, performing overdamped motion under drag effects, due to a rotating resisting medium. It is related to models that have been recently applied to the study of type-II superconductors. The relevance of the present developments to the study of complex systems in physics, astronomy, and biology, is discussed.