Non-extensive random walks

Details Non-extensive random walks Non-extensive random walks

2004-09-01

arxiv.org/abs/cond-mat/0409035v1

Physics

Condensed Matter

Statistical Mechanics

5 pages, 2 figures

Scientific paper

10.1016/j.physa.2005.06.052

The stochastic properties of variables whose addition leads to $q$-Gaussian distributions $G_q(x)=[1+(q-1)x^2]_+^{1/(1-q)}$ (with $q\in\mathbb{R}$ and where $[f(x)]_+=max\{f(x),0\}$) as limit law for a large number of terms are investigated. These distributions have special relevance within the framework of non-extensive statistical mechanics, a generalization of the standard Boltzmann-Gibbs formalism, introduced by Tsallis over one decade ago. Therefore, the present findings may have important consequences for a deeper understanding of the domain of applicability of such generalization. Basically, it is shown that the random walk sequences, that are relevant to this problem, possess a simple additive-multiplicative structure commonly found in many contexts, thus justifying the ubiquity of those distributions. Furthermore, a connection is established between such sequences and the nonlinear diffusion equation $\partial_t \rho=\partial^2_{xx}\rho^\nu$ ($\nu\neq1$).

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