The world according to Renyi: Thermodynamics of multifractal systems

2002/07/31 by Petr Jizba, Toshihico Arimitsu · 2 citations
Physics and Astronomy · Mathematics · #cond-mat.stat-mech #astro-ph #hep-th #math-ph #math.MP #physics.data-an

paper · pdf · doi:10.1016/j.aop.2004.01.002

published as Annals Phys. 312 (2004) 17-57 · 24 pages, 2 figures, RevTex, thoroughly revised, accepted to Annals of Physics

arxiv created 2003/12/26 · arxiv updated 2009/11/30

Abstract

We discuss basic statistical properties of systems with multifractal structure. This is possible by extending the notion of the usual Gibbs--Shannon entropy into more general framework - Renyi's information entropy. We address the renormalization issue for Renyi's entropy on (multi)fractal sets and consequently show how Renyi's parameter is connected with multifractal singularity spectrum. The maximal entropy approach then provides a passage between Renyi's information entropy and thermodynamics on multifractals. Important issues as, for instance, Renyi's entropy versus Tsallis--Havrda--Charvat entropy and PDF reconstruction theorem are also studied. Finally, some further speculations on a possible relevance of our approach to cosmology are discussed.

Cited by

• Rényi and Tsallis information entropies for the Darboux III quantum nonlinear oscillator
• Through and beyond moments, entropies and Fisher information measures: new informational functionals and inequalities

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