caa a22 4500 467893373 CHVBK 20180406152755.0 cr unu---uuuuu 170328e20061001xx s 000 0 eng 10.1007/s10955-006-9174-z doi (NATIONALLICENCE)springer-10.1007/s10955-006-9174-z Nonconcave Entropies in Multifractals and the Thermodynamic Formalism [Elektronische Daten] [Hugo Touchette, Christian Beck] 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. Springer Science + Business Media, Inc., 2006 multifractals nationallicence multifractal spectrum nationallicence entropy nationallicence free energy nationallicence Legendre transform nationallicence thermodynamic formalism nationallicence Touchette Hugo School of Mathematical Sciences, Queen Mary, University of London, E1 4NS, London, UK aut Beck Christian School of Mathematical Sciences, Queen Mary, University of London, E1 4NS, London, UK aut Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 125/2(2006-10-01), 455-471 0022-4715 125:2<455 2006 125 10955 https://doi.org/10.1007/s10955-006-9174-z text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10955-006-9174-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Touchette Hugo School of Mathematical Sciences, Queen Mary, University of London, E1 4NS, London, UK aut NATIONALLICENCE 700 1- Beck Christian School of Mathematical Sciences, Queen Mary, University of London, E1 4NS, London, UK aut NATIONALLICENCE 773 0- Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 125/2(2006-10-01), 455-471 0022-4715 125:2<455 2006 125 10955 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer