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• Essex, C
• Hoffmann, K
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Entropy 2012, 14(4), 701-716; doi:10.3390/e14040701

Article

Tsallis Relative Entropy and Anomalous Diffusion

Janett Prehl ¹ , Christopher Essex ²  and Karl Heinz Hoffmann ^1,*

¹ Institute of Physics, Chemnitz University of Technology, D-09107 Chemnitz, Germany ² Department of Applied Mathematics, University of Western Ontario, Middlesex College, London, ON, N6A 5B7, Canada

* Author to whom correspondence should be addressed.

Received: 1 March 2012; in revised form: 19 March 2012 / Accepted: 30 March 2012 / Published: 10 April 2012

(This article belongs to the Special Issue Tsallis Entropy)

Download PDF Full-Text [182 KB, uploaded 11 April 2012 09:13 CEST]

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

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