Nonextensive statistical mechanics: a brief review of its present status

Abstract

We briefly review the present status of nonextensive statistical mechanics. We focus on (i) the central equations of the formalism, (ii) the most recent applications in physics and other sciences, (iii) the a priori determination (from microscopic dynamics) of the entropic index q for two important classes of physical systems, namely low-dimensional maps (both dissipative and conservative) and long-range interacting many-body hamiltonian classical systems.

Determining Dynamical Path Distributions usingMaximum Relative Entropy

“…It merely restricts the form of the entropy to a one-dimensional continuum labeled by a parameter . The resulting -entropies are equivalent to those proposed by Renyi [12] or by Tsallis [13] in the sense that they lead to the same updated probabilities.…”

q-Deformed Loewner Evolution

“…In particular, a form of q-calculus is involved in the description of non-extensive statistical mechanics [34], whose scope of application ranges from phenomena where the Loewner approach has already proved useful, like the Ising model [35] and viscous fingering [36], to broader and more general subjects, as generalizations of the α-stable distributions [37] and anomalous diffusion [38]. We do not pursue these connections in this paper, but they constitute the physical motivations for a systematic approach to q-deformed equations in statistical mechanics.…”

Critical Behavior of Systems With Long-Range Interaction in Restricted Geometry

“…The opposite case, corresponding to a LR interaction with −d ≤ σ ≤ 0, sometimes called also nonintegrable interaction, has been studied in connection with the so called "nonextensive thermodynamics" (see [16] and references therein). In spite of the recent growing interest, the case of nonintegrable interaction is beyond the scope of the present study.…”