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.
Highlights
Nonlinear kinetics in the coordinate or velocity space or in the phase space has been used systematically over the last few decades
The derivations in the preceding section show how the nonlinear Fokker–Planck equation can be obtained from the discrete Fokker–Planck current of Equation (23), which is derived from the simple, Kinetic Interaction Principle (KIP)-based master Equation (7)
The KIP-based master equation in the continuum limit yielded the already known nonlinear-Fokker–Planck Equation (2), which is widely used in the literature to describe anomalous diffusion in condensed matter physics and nonconventional statistical physics
Summary
Nonlinear kinetics in the coordinate or velocity space or in the phase space has been used systematically over the last few decades. [1], the most general nonlinear kinetics in phase space was considered in the limit of the diffusive approximation In this limit, the interaction between the particles and the medium is viewed as a diffusive process in the velocity space. It is important to note that the right hand side of Equation (1), which introduces nonlinear effects into kinetics, describes an unconventional diffusion process in the velocity space. This process can clearly be studied in the coordinate space, where it describes an anomalous diffusion process. The present paper shows that the Kinetic Interaction Principle (KIP) which governs the particle kinetics [1] univocally defines a very simple lattice-based master equation, which yields the nonlinear.
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