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Entropy 2013, 15(8), 2989-3006; doi:10.3390/e15082989

Time Evolution of Relative Entropies for Anomalous Diffusion

Institut für Physik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany
Department of Applied Mathematics, The University of Western Ontario, N6A 5B7 London, Canada
Author to whom correspondence should be addressed.
Received: 26 June 2013 / Revised: 17 July 2013 / Accepted: 18 July 2013 / Published: 26 July 2013
(This article belongs to the Special Issue Distance in Information and Statistical Physics Volume 2)
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The entropy production paradox for anomalous diffusion processes describes a phenomenon where one-parameter families of dynamical equations, falling between the diffusion and wave equations, have entropy production rates (Shannon, Tsallis or Renyi) that increase toward the wave equation limit unexpectedly. Moreover, also surprisingly, the entropy does not order the bridging regime between diffusion and waves at all. However, it has been found that relative entropies, with an appropriately chosen reference distribution, do. Relative entropies, thus, provide a physically sensible way of setting which process is “nearer” to pure diffusion than another, placing pure wave propagation, desirably, “furthest” from pure diffusion. We examine here the time behavior of the relative entropies under the evolution dynamics of the underlying one-parameter family of dynamical equations based on space-fractional derivatives.
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 (CC BY 3.0).

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MDPI and ACS Style

Prehl, J.; Boldt, F.; Essex, C.; Hoffmann, K.H. Time Evolution of Relative Entropies for Anomalous Diffusion. Entropy 2013, 15, 2989-3006.

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