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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


On the Way to Generalized Kinetics for Multi-scale Complex System

Discontinuity, Nonlinearity, and Complexity 8(2) (2019) 199--210 | DOI:10.5890/DNC.2019.06.007

V. V. Uchaikin

Ulyanovsk State University, Ulyanovsk, Rissia

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Development of nano-technology throws a new challenge to statistical physics. The classical statistical physics is underlaid by classical or quantum mechanics of closed systems, classical version of the Central Limit Theorem, the Markov model of Brownian motion, the linear approximation of the perturbation theory as a main mathematical tool. The nanotechnology has caused constructing new models: nonlocal in space and time dynamics, generalized Limit Theorem for the Levy-stable family as limit distributions, non-Markovian (subordinated) Levy-motion and instead of the Brownian one, self-similar (fractal) and non-local (integral and fractional differential) analysis. The article gives a short review of these concepts. The multiscale complex system is considered to exacute a selfsimilar motion obeying the Levy stable statistics, which generalized then to fractionally stable kind. The intermediate asymptotics is introduced on the base of truncated Lévy flight. This leads to the correspondence principle connecting classical and nano-statistics. Some numerical results are presented.


This work is partially supported by the Russian Foundation of Basic Research (Project 18-51- 53018).


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