Non-linear kinetics underlying generalized statistics
By Giorgio Kaniadakis. 2001
494 citations recorded by the source corpus.
The purpose of the present effort is threefold. Firstly, it is shown that there exists a principle, that we call kinetical interaction principle (KIP), underlying the non-linear kinetics in particle systems, independently on the picture (Kramers, Boltzmann) used to describe their time evolution. Secondly, the KIP imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution, both as stationary solution of the non-linear evolution equation and as the state which maximizes the generalized entropy. Thirdly, the KIP allows, on one hand, to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and, on the other hand, suggests how we can introduce naturally new distributions. Finally, as a working example of the approach to the non-linear kinetics here presented, a new non-extensive statistics is constructed and studied starting from a one-parameter deformation of the exponential function holding the relation f(−x)f(x)=1.
Subfields: Economics and Econometrics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/fa8f0cee06758b67b7c8feabd5875a9c5f613706
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Citations in the corpus
- A new model of income distribution: the κ-generalized distribution (2011, unknown): “In this paper we introduce a new three-parameter distribution based on the generalization of the exponential and logarithm functions proposed by Kaniadakis (2001, 2002, 2005) and defined as”
- Pareto's Law of Income Distribution: Evidence for Germany, the United Kingdom, and the United States (2005, methodology): “When fits are performed, a two-parameter lognormal distribution is used for the low-middle 1 Recently, a distribution proposed by [14,15] has the form of a deformed exponential function: Pκ (x) = ( √ 1 + κ2x2 − κx ) 1 κ”
- Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations (2007, unknown): “The appropriate thermodynamical potential is the free energy F = E − TS which decreases monotonically with time at fixed mass. Generalized Boltzmann and Landau equations describing conservative systems where the energy is fixed have been introduced by Kaniadakis (2001) [12] and Chavanis (2004) [34].”
- Entropy and equilibrium via games of complexity (2004, result): “It is shown that the entropy measures of Tsallis and Kaniadakis can be discussed from this point of view by embedding them in one and the same two-parameter family of entropy functions (and associated complexity measures).”
- Deformed exponentials and logarithms in generalized thermostatistics (2002, background): “On the otherhand, using Kaniadakis’de nitionsoneobtains pk = Ek + q 1+ ( Ek) 2 1= : (40) In thiscasetheprobabilitiespk arealwaysstrictly positive.”