Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle
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 f...
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MDPI AG
2018-06-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/20/6/426 |
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| author | Giorgio Kaniadakis Dionissios T. Hristopulos |
| author_facet | Giorgio Kaniadakis Dionissios T. Hristopulos |
| author_sort | Giorgio Kaniadakis |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-12T19:40:28Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-12T19:40:28Z |
| publishDate | 2018-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-05bfeaf235c644d6aaaf08ebc750d0922022-12-22T03:19:05ZengMDPI AGEntropy1099-43002018-06-0120642610.3390/e20060426e20060426Nonlinear Kinetics on Lattices Based on the Kinetic Interaction PrincipleGiorgio Kaniadakis0Dionissios T. Hristopulos1Department of Applied Science and Technology, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, ItalySchool of Mineral Resources Engineering, Technical University of Crete, 73100 Chania, GreeceMaster 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.http://www.mdpi.com/1099-4300/20/6/426Fokker–Planck equationsfermion statisticsboson statisticsHaldane statisticsKinetic interaction principleanomalous diffusionFokker–Planck current |
| spellingShingle | Giorgio Kaniadakis Dionissios T. Hristopulos Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle Entropy Fokker–Planck equations fermion statistics boson statistics Haldane statistics Kinetic interaction principle anomalous diffusion Fokker–Planck current |
| title | Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle |
| title_full | Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle |
| title_fullStr | Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle |
| title_full_unstemmed | Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle |
| title_short | Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle |
| title_sort | nonlinear kinetics on lattices based on the kinetic interaction principle |
| topic | Fokker–Planck equations fermion statistics boson statistics Haldane statistics Kinetic interaction principle anomalous diffusion Fokker–Planck current |
| url | http://www.mdpi.com/1099-4300/20/6/426 |
| work_keys_str_mv | AT giorgiokaniadakis nonlinearkineticsonlatticesbasedonthekineticinteractionprinciple AT dionissiosthristopulos nonlinearkineticsonlatticesbasedonthekineticinteractionprinciple |