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Entropy 2017, 19(4), 155; doi:10.3390/e19040155

Random Walks Associated with Nonlinear Fokker–Planck Equations

1
Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, Maringá 87020-900, Brazil
2
National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil
3
Departamento de Física, Universidade Estadual de Ponta Grossa, Av. General Carlos Cavalcanti 4748, Ponta Grossa 84030-900, Brazil
*
Author to whom correspondence should be addressed.
Academic Editor: Angelo Plastino
Received: 24 February 2017 / Revised: 28 March 2017 / Accepted: 30 March 2017 / Published: 1 April 2017
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Abstract

A nonlinear random walk related to the porous medium equation (nonlinear Fokker–Planck equation) is investigated. This random walk is such that when the number of steps is sufficiently large, the probability of finding the walker in a certain position after taking a determined number of steps approximates to a q-Gaussian distribution ( G q , β ( x ) [ 1 ( 1 q ) β x 2 ] 1 / ( 1 q ) ), which is a solution of the porous medium equation. This can be seen as a verification of a generalized central limit theorem where the attractor is a q-Gaussian distribution, reducing to the Gaussian one when the linearity is recovered ( q 1 ). In addition, motivated by this random walk, a nonlinear Markov chain is suggested. View Full-Text
Keywords: anomalous diffusion; random walks; long-tailed distributions; Markov chains anomalous diffusion; random walks; long-tailed distributions; Markov chains
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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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dos Santos Mendes, R.; Lenzi, E.K.; Malacarne, L.C.; Picoli, S.; Jauregui, M. Random Walks Associated with Nonlinear Fokker–Planck Equations. Entropy 2017, 19, 155.

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