42 research outputs found
Nonconcave entropies in multifractals and the thermodynamic formalism
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.Comment: 9 pages, revtex4, 3 figures. Changes in v2: sections added on
applications plus many new references; contains an addendum not contained in
published versio
A continuity property of the convolution
It is shown that Pn --> P0 weakly (i.e. [latin small letter f with hook][latin small letter f with hook]dPn --> [latin small letter f with hook][latin small letter f with hook]dP0 for every continuous and bounded function [latin small letter f with hook] defined on ) for probability measures Pn, n[epsilon][union or logical sum]{0}, on the Borel subssets of , implies in the sense of Brooks and Chacon for any probability measure P on the Borel subsets of .convolution [omega]2-convergence