Competing Styles of Statistical Mechanics: I. Systematization and Clarification in a General Theory

TL;DR

This paper clarifies and systematizes competing styles of statistical mechanics that extend conventional Boltzmann-Gibbs theory using unconventional approaches like Renyi's statistics, especially for systems with fractal or scaling properties.

Contribution

It provides a detailed description and clarification of alternative statistical mechanics approaches, including the construction of a nonequilibrium ensemble formalism based on Renyi's statistics.

Findings

Unconventional distribution functions for fermions and bosons derived.

Framework applicable to systems with fractal and scaling features.

Foundation for future applications in nanometric and fractal-like systems.

Abstract

Competing styles of Statistical Mechanics have been introduced as practical succedaneous to the conventional well established Boltzmann-Gibbs statistical mechanics, when in the use of the latter the researcher is impaired in his/her capacity for satisfying the Criteria of Efficiency and/or Sufficiency in statistics [Fisher, 1922], that is, a failure in the characterization (presence of fractality, scaling, etc.) of the system related to some aspect relevant to the given physical situation. To patch this limitation on the part of the observer, in order to make predictions on the values of observables and response functions, are introduced unconventional approaches. We present a detailed description of their construction and a clarification of its scope and interpretation. Also, resorting to the use of the particular case of Renyi's unconventional statistics is built a nonequilibrium ensemble formalism. The unconventional distribution functions of fermions and bosons are obtained, and in a follow-up article [cond-mat/0306247] we describe applications to the study of experimental results in semiconductor physics and in electro-chemistry involving nanometric scales and fractal-like structures, and some additional theoretical analysis is added. PACS: 05.70.Ln, 82.20.Mj, 82.20.Db Keywords: Nonequilibrium Ensemble Formalism; Generalized Informational Entropies; Generalized Statistics; Nonextensive Statistics; Renyi Statistics; Escort Probability.