Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle

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.

Classical Model of Quons

“…Differently, within the second approaches we find: Haldane-Wu statistics [34,35], including semion [36], obtained by generalizing Pauli exclusion principle; intermediate statistics by Gentile [37], derived in a thermodynamical context by assuming that the maximum occupation particle number of an energy level is between one (standard fermions) and infinity (standard bosons); and more recently the interpolating boson-fermion statistics [38][39][40][41][42], used to study non relativistic quantum systems that obey to a generalized inclusion/exclusion principle, obtained by modifying the dependency of the transition probability from the occupation particle number of the starting and the arriving site [43,44].…”

The Longitudinal Plasma Modes of κ-Deformed Kaniadakis Distributed Plasmas Carrying Orbital Angular Momentum

“…In this context, Rényi proposed the non-extensive generalization of BG entropy [ 34 ], which was later also suggested by Tsallis [ 35 ], whose non-extensive entropy was in excellent agreement [ 36 ] with experimental data [ 37 , 38 ]. Afterwards, Kaniadakis put forward a new so-called -deformed distribution in 2001 [ 39 ], which has been widely applied to the kinetics of interaction atoms and photons [ 40 ], nonlinear kinetics [ 41 , 42 , 43 ], cosmic rays [ 33 ], blackbody radiation [ 44 ], quantum entanglement [ 45 ], quark–gluon plasma formation [ 46 ], and even financial systems [ 47 , 48 ] and epidemiology [ 49 ]. The -deformed distribution arising from Kaniadakis entropy covers both nonextensive and the classical Maxwell–Boltzman distributions [ 39 ].…”