Tsallis Relative Entropy and Anomalous Diffusion
Open Access
- 10 April 2012
- Vol. 14 (4), 701-716
- https://doi.org/10.3390/e14040701
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.This publication has 40 references indexed in Scilit:
- Search researchNature, 2006
- Optimal Search Strategies for Hidden TargetsPhysical Review Letters, 2005
- Kullback-leibler approximation of spectral density functionsIEEE Transactions on Information Theory, 2003
- Nonlinear equation for anomalous diffusion: Unified power-law and stretched exponential exact solutionPhysical Review E, 2001
- The similarity group and anomalous diffusion equationsJournal of Physics A: General Physics, 2000
- Testing exponentiality of the residual life, based on dynamic Kullback-Leibler informationIEEE Transactions on Reliability, 1998
- Fractional Diffusion and Entropy ProductionJournal of Non-Equilibrium Thermodynamics, 1998
- Fractional diffusion and wave equationsJournal of Mathematical Physics, 1989
- Diffusion in disordered mediaAdvances in Physics, 1987
- On Information and SufficiencyThe Annals of Mathematical Statistics, 1951