Nonextensive statistical mechanics: A brief introduction

Details Nonextensive statistical mechanics: A brief introduction Nonextensive statistical mechanics: A brief introduction

2003-05-26

arxiv.org/abs/cond-mat/0305606v1

Continuum Mech. Thermodyn. 16, 223-235 (2004)

Physics

Condensed Matter

Statistical Mechanics

Invited paper to appear in "Extensive and non-extensive entropy and statistical mechanics", a topical issue of Continuum Mecha

Scientific paper

10.1007/s00161-004-0174-4

Boltzmann-Gibbs statistical mechanics is based on the entropy $S_{BG}=-k \sum_{i=1}^W p_i \ln p_i$. It enables a successful thermal approach of ubiquitous systems, such as those involving short-range interactions, markovian processes, and, generally speaking, those systems whose dynamical occupancy of phase space tends to be ergodic. For systems whose microscopic dynamics is more complex, it is natural to expect that the dynamical occupancy of phase space will have a less trivial structure, for example a (multi)fractal or hierarchical geometry. The question naturally arises whether it is possible to study such systems with concepts and methods similar to those of standard statistical mechanics. The answer appears to be {\it yes} for ubiquitous systems, but the concept of entropy needs to be adequately generalized. Some classes of such systems can be satisfactorily approached with the entropy $S_q=k\frac{1-\sum_{i=1}^W p_i^q}{q-1}$ (with $q \in \cal R$, and $S_1 =S_{BG}$). This theory is sometimes referred in the literature as {\it nonextensive statistical mechanics}. We provide here a brief introduction to the formalism, its dynamical foundations, and some illustrative applications. In addition to these, we illustrate with a few examples the concept of {\it stability} (or {\it experimental robustness}) introduced by B. Lesche in 1982 and recently revisited by S. Abe.

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