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.
Highlights
In the beginning of the last century, important results concerning Brownian motion were obtained by Einstein [1], Langevin [2], Fokker [3], and Planck [4], among others
We have considered a random walk that arises from a nonlinear Fokker–Planck equation and some of its consequences
The random walk analyzed here generalizes the simplest one via a nonlinear Markov chain obtained from the porous media equation
Summary
In the beginning of the last century, important results concerning Brownian motion were obtained by Einstein [1], Langevin [2], Fokker [3], and Planck [4], among others. Starting from a random walk (discrete case) with steps that are independent and of equal length, we can obtain a normal diffusion process (continuous case) when we let the number of steps increase without bound This diffusion process is governed by a linear Fokker–Planck equation, whose solution is the Gaussian distribution. This random walk will show strong correlations since—as we will show—the distribution of the position of the walker after taking a large number of steps will approximate the solution of the nonlinear Fokker–Planck equation, which is not the Gaussian distribution In this direction, we will verify a kind of generalization of the central limit theorem.
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