The world according to Renyi: Thermodynamics of multifractal systems

Details The world according to Renyi: Thermodynamics of multifractal systems The world according to Renyi: Thermodynamics of multifractal systems

2002-07-30

arxiv.org/abs/cond-mat/0207707v3

Annals Phys. 312 (2004) 17-57

Physics

Condensed Matter

Statistical Mechanics

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

Scientific paper

10.1016/j.aop.2004.01.002

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.

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