A problem with the escort distribution representation of nonextensive statistical mechanics.pdf

a r X i v : c o n d - m a t / 0 0 0 6 0 5 3 4 J u n 2 0 0 0 1 A problem with the escort distribution representation of nonextensive statistical mechanics Sumiyoshi Abe College of Science and Technology, Nihon University, Funabashi, Chiba 274-8501, Japan It is pointed out that the Tsallis entropy functional represented in terms of the escort distribution is not concave if the entropic index q is less than unity. It is emphasized that the escort distribution is a secondary object calculated from the basic original distribution. PACS number: 05.20.-y 2In recent years, there has been growing interest in a nonextensive generalization of Boltzmann-Gibbs statistical mechanics. This formalism termed nonextensive statistical mechanics was initiated by Tsallis [1] by introducing a generalized entropy and was applied to various physical systems with long-range interactions, long-range memories, (multi)fractal structure and spatio-temporal complexity 1 . Among others, its recent impressive success in describing the phenomena of fully developed turbulence should be noticed [2,3]. Nonextensive statistical mechanics has a specific mathematical structure. It makes use of the concept of the escort distribution [4] to define expectation values of physical quantities. Since the escort distribution plays an essential role, some manipulations have been made to represent the theory completely in terms of it in the literature (which we do not specify here). The purpose of this short note is to call an attention to a problem regarding the concavity property of the entropy functional in the escort distribution representation. The escort distribution cannot be taken as a basic variable in maximum entropy calculus and is a secondary object calculated from the basic original distribution. To make the point clear, it seems appropriate to first summarize the standard formalism of nonextensive statistical mechanics. This theory is based on the Tsallis