Nonlinear Kinetics on Lattices based on the Kinetic Interaction Principle

TL;DR

This paper demonstrates that the Kinetic Interaction Principle (KIP) uniquely determines a master equation that accurately models nonlinear kinetics on lattices, ensuring correct discretization of nonlinear Fokker-Planck equations in physical systems.

Contribution

It establishes that KIP provides a unique, physically motivated master equation for nonlinear lattice kinetics, bridging discrete models and continuum nonlinear Fokker-Planck equations.

Findings

KIP defines a simple, unique master equation for nonlinear lattice kinetics.

The master equation converges to the most general nonlinear Fokker-Planck equation in the continuum limit.

Provides a physically consistent discretization scheme for nonlinear Fokker-Planck equations.

Abstract

Master equations define the dynamics that govern the time evolution of various physical processes on lattices. In the continuum limit, master equations lead to Fokker-Planck partial differential equations that represent the dynamics of physical systems in continuous spaces. Over the last few decades, nonlinear Fokker-Planck equations have become very popular in condensed matter physics and in statistical physics. Numerical solutions of these equations require the use of discretization schemes. However, the discrete evolution equation obtained by the discretization of a Fokker-Planck partial differential equation depends on the specific discretization scheme. In general, the discretized form is different from the master equation that has generated the respective Fokker-Planck equation in the continuum limit. Therefore, the knowledge of the master equation associated with a given Fokker-Planck equation is extremely important for the correct numerical integration of the latter, since it provides a unique, physically motivated discretization scheme. This paper shows that the Kinetic Interaction Principle (KIP) that governs the particle kinetics of many body systems, introduced in [G. Kaniadakis, Physica A, 296, 405 (2001)], univocally defines a very simple master equation that in the continuum limit yields the nonlinear Fokker-Planck equation in its most general form.