Generalized Fokker-Planck equations and effective thermodynamics

TL;DR

This paper introduces a new class of generalized Fokker-Planck equations linked to an effective thermodynamics framework, capable of modeling complex phenomena like anomalous diffusion and phase transitions in interacting particle systems.

Contribution

It develops a novel class of Fokker-Planck equations that unify various physical phenomena and connect to phase transition theory, extending the classical models.

Findings

Equations describe anomalous diffusion and quantum statistics.

Long-range interactions lead to phase transitions and blow-up phenomena.

Short-range limits recover Cahn-Hilliard equations.

Abstract

We introduce a new class of Fokker-Planck equations associated with an effective generalized thermodynamical framework. These equations describe a gas of Langevin particles in interaction. The free energy can take various forms which can account for anomalous diffusion, quantum statistics, lattice models... When the potential of interaction is long-ranged, these equations display a rich structure associated with canonical phase transitions and blow-up phenomena. In the limit of short-range interactions, they reduce to Cahn-Hilliard equations.