Nonconcave entropies in multifractals and the thermodynamic formalism

TL;DR

This paper examines the limitations of Legendre-Fenchel transforms in accurately deriving multifractal spectra, especially when the spectra are nonconcave, and proposes a generalized free energy approach to address this issue.

Contribution

It reveals the conditions under which Legendre-Fenchel transforms fail for nonconcave spectra and introduces a generalized free energy method to overcome this limitation.

Findings

Legendre-Fenchel transform yields the spectrum's concave envelope if the spectrum is nonconcave.

Nonconcave multifractal spectra cannot be directly obtained from standard free energy functions.

A generalized free energy approach can accurately recover nonconcave spectra.

Abstract

We discuss a subtlety involved in the calculation of multifractal spectra when these are expressed as Legendre-Fenchel transforms of functions analogous to free energy functions. We show that the Legendre-Fenchel transform of a free energy function yields the correct multifractal spectrum only when the latter is wholly concave. If the spectrum has no definite concavity, then the transform yields the concave envelope of the spectrum rather than the spectrum itself. Some mathematical and physical examples are given to illustrate this result, which lies at the root of the nonequivalence of the microcanonical and canonical ensembles. On a more positive note, we also show that the impossibility of expressing nonconcave multifractal spectra through Legendre-Fenchel transforms of free energies can be circumvented with the help of a generalized free energy function, which relates to a recently introduced generalized canonical ensemble. Analogies with the calculation of rate functions in large deviation theory are finally discussed.