The world according to Renyi: Thermodynamics of multifractal systems

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.

arXiv comment24 pages, 2 figures, RevTex, thoroughly revised, accepted to Annals of Physics

References in corpus (6)

• Cosmological Experiments in Condensed Matter Systems
• Integrable Structure of Interface Dynamics
• Axioms and uniqueness theorem for Tsallis entropy
• Measure representation and multifractal analysis of complete genomes
• PDF of Velocity Fluctuation in Turbulence by a Statistics based on Generalized Entropy
• Pressure of the Non-equilibrium O(N) Phi^{4} Theory in the Large N Limit

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