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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; in revised form: 17 July 2013 / Accepted: 18 July 2013 / Published: 26 July 2013
Abstract: 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
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Prehl, J.; Boldt, F.; Essex, C.; Hoffmann, K.H. Time Evolution of Relative Entropies for Anomalous Diffusion. Entropy 2013, 15, 2989-3006.
Prehl J, Boldt F, Essex C, Hoffmann KH. Time Evolution of Relative Entropies for Anomalous Diffusion. Entropy. 2013; 15(8):2989-3006.
Prehl, Janett; Boldt, Frank; Essex, Christopher; Hoffmann, Karl H. 2013. "Time Evolution of Relative Entropies for Anomalous Diffusion." Entropy 15, no. 8: 2989-3006.