It is a rather common belief that the only probability distribution occurring in the statistical physics of many-particle systems is that of Boltzmann and Gibbs (BG). This point of view is too limited. The BG-distribution, when seen as a function of parameters such as the inverse temperature and the chemical potential, is a member of the exponential family. This observation is important to understand the structure of statistical mechanics and its connection with thermodynamics. It also is the starting point of the generalizations discussed below. Recently, the notion of a generalized exponential family has been introduced, both in the mathematics and in the physics literature. A sub-class of this generalized family is the q-exponential family, where q is a real parameter describing the deformation of the exponential function. It is the intention of this talk to show the relevance for statistical physics of these generalizations of the BG- distribution. Particular attention will go to the configurational density of classical mono-atomic gases in the micro- canonical ensemble. These belong to the q-exponential family, where q tends to 1 as the number of particles tends to infinity. Hence, in this limit the density converges to the BG-distribution.
Mathematical Aspects of Generalized Entropies and their Applications
(University of Antwerp)
joint with DSSP seminar
Thu, 05/11/2009 - 17:00