# A General Nonlinear Fokker-Planck Equation and its Associated Entropy

## TL;DR

This paper explores a broad nonlinear Fokker-Planck equation derived from master equations, analyzing its stationary states and introducing generalized entropies that satisfy the H-theorem, linking transition rates to entropy parameters.

## Contribution

It provides a comprehensive analysis of a general nonlinear Fokker-Planck equation, deriving stationary solutions and formulating associated generalized entropies with constraints.

## Key findings

- Derived stationary-state distributions for special cases.
- Introduced classes of generalized entropies satisfying the H-theorem.
- Linked transition rate parameters to entropy parameters with specific restrictions.

## Abstract

A recently introduced nonlinear Fokker-Planck equation, derived directly from a master equation, comes out as a very general tool to describe phenomenologically systems presenting complex behavior, like anomalous diffusion, in the presence of external forces. Such an equation is characterized by a nonlinear diffusion term that may present, in general, two distinct powers of the probability distribution. Herein, we calculate the stationary-state distributions of this equation in some special cases, and introduce associated classes of generalized entropies in order to satisfy the H-theorem. Within this approach, the parameters associated with the transition rates of the original master-equation are related to such generalized entropies, and are shown to obey some restrictions. Some particular cases are discussed.