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Special Issue: Tsallis Entropy
Entropy 2012, 14(4), 701-716; doi:10.3390/e14040701
Article

Tsallis Relative Entropy and Anomalous Diffusion

1
,
2
 and
1,*
Received: 1 March 2012 / Revised: 19 March 2012 / Accepted: 30 March 2012 / Published: 10 April 2012
(This article belongs to the Special Issue Tsallis Entropy)
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Abstract

In this paper we utilize the Tsallis relative entropy, a generalization of the Kullback–Leibler entropy in the frame work of non-extensive thermodynamics to analyze the properties of anomalous diffusion processes. Anomalous (super-) diffusive behavior can be described by fractional diffusion equations, where the second order space derivative is extended to fractional order α ∈ (1, 2). They represent a bridging regime, where for α = 2 one obtains the diffusion equation and for α = 1 the (half) wave equation is given. These fractional diffusion equations are solved by so-called stable distributions, which exhibit heavy tails and skewness. In contrast to the Shannon or Tsallis entropy of these distributions, the Kullback and Tsallis relative entropy, relative to the pure diffusion case, induce a natural ordering of the stable distributions consistent with the ordering implied by the pure diffusion and wave limits.
Keywords: space-fractional diffusion equation; stable distribution; Kullback–Leibler entropy; Tsallis relative entropy space-fractional diffusion equation; stable distribution; Kullback–Leibler entropy; Tsallis relative entropy
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.

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Prehl, J.; Essex, C.; Hoffmann, K.H. Tsallis Relative Entropy and Anomalous Diffusion. Entropy 2012, 14, 701-716.

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