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Entropy 2018, 20(6), 426; https://doi.org/10.3390/e20060426

Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle

1
Department of Applied Science and Technology, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
2
School of Mineral Resources Engineering, Technical University of Crete, 73100 Chania, Greece
3
Telecommunication Systems Institute, 73100 Chania, Greece
*
Author to whom correspondence should be addressed.
Received: 30 April 2018 / Revised: 26 May 2018 / Accepted: 30 May 2018 / Published: 1 June 2018
(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)
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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. View Full-Text
Keywords: Fokker–Planck equations; fermion statistics; boson statistics; Haldane statistics; Kinetic interaction principle; anomalous diffusion; Fokker–Planck current Fokker–Planck equations; fermion statistics; boson statistics; Haldane statistics; Kinetic interaction principle; anomalous diffusion; Fokker–Planck current
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Kaniadakis, G.; Hristopulos, D.T. Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle. Entropy 2018, 20, 426.

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