Nonconcave entropies in multifractals and the thermodynamic formalism

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.

9 pages, revtex4, 3 figures. Changes in v2: sections added on applications plus many new references; contains an addendum not contained in published version

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• Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume-Emery-Griffiths model
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• Generalized canonical ensembles and ensemble equivalence
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Cited by in corpus (6)

• The large deviation approach to statistical mechanics
• Generic Multifractality in Exponentials of Long Memory Processes
• Negative magnetic susceptibility and nonequivalent ensembles for the mean-field ${\varphi }^4$ spin model
• Multifractal spectrum of phase space related to generalized thermostatistics
• Large-scale fluctuations of the largest Lyapunov exponent in diffusive systems
• Specific behavior of one chaotic dynamics near the fine-structure constant