The world according to Renyi: Thermodynamics of multifractal systems

TL;DR

This paper explores the thermodynamics of multifractal systems using Renyi's entropy, establishing connections between entropy, multifractal spectra, and potential cosmological implications.

Contribution

It extends the concept of entropy to multifractal systems, linking Renyi's parameter with the multifractal spectrum and bridging thermodynamics with multifractal analysis.

Findings

Renyi's entropy relates to the multifractal singularity spectrum.

A maximal entropy approach connects Renyi's entropy with thermodynamics.

Discussion of Renyi versus Tsallis entropy and implications for cosmology.

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.