The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks

Abstract

The nonadditive entropy S q has been introduced in 1988 focusing on a generalization of Boltzmann-Gibbs (BG) statistical mechanics. The aim was to cover a (possibly wide) class of systems among those very many which violate hypothesis such as ergodicity, under which the BG theory is expected to be valid. It is now known that S q has a large applicability; more specifically speaking, even outside Hamiltonian systems and their thermodynamical approach. In the present paper we review and comment some relevant aspects of this entropy, namely (i) Additivity versus extensivity; (ii) Probability distributions that constitute attractors in the sense of Central Limit Theorems; (iii) The analysis of paradigmatic low-dimensional nonlinear dynamical systems near the edge of chaos; and (iv) The analysis of paradigmatic long-range-interacting many-body classical Hamiltonian systems. Finally, we exhibit recent as well as typical predictions, verifications and applications of these concepts in natural, artificial, and social systems, as shown through theoretical, experimental, observational and computational results.

Dynamic properties of small-scale solar wind plasma fluctuations

“…[57,58]). Nonlinear interactions can create fractal structuring of the phase space and produce global correlations in the entire multiscale system.…”

Nonextensive Dynamics of Drifting Sea Ice

“…e thermodynamic description of processes in nonequilibrium statistical systems was developed by Tsallis [3,4]. e main idea forwarded in his approach was to introduce into additive expression (5) a cross-member that would take into account some interactions between subsystems through a "parameter of nonextensivity" :…”

“…Passing over details of the Tsallis' consideration (see [3,4] for details), we note that the classical statistical mechanics responds the limit ((6) transforms into (5)); for the formalism imposes a limited number of events in equilibrium system; �nally, the value indicates the nonequilibrium state of the dynamic system-this is the case of non-extensive dynamics, which is incompatible with the entropic additivity.…”

“…In recent years, a new statistical concept called the nonextensive statistical mechanics (NESM) was developed by Tsallis [3,4] for thermodynamic description of the behavior of multiscale systems revealing �uctuations around their equilibrium state. e NESM was successively applied for assessing the degree of deviation of natural dynamic systems from their equilibrium state prior to large-scale hazards, such as earthquakes [5,6] and �oods [7].…”

Nonlinear Wave Equations Related to Nonextensive Thermostatistics

“…In the present paper, we introduce and investigate some features of two nonlinear wave equations related to the nonextensive thermostatistical formalism [1][2][3]. This work is motivated by a recent line of enquiry concerning nonlinear extensions of the Schroedinger, Dirac, and Klein-Gordon equations also linked to this formalism [4][5][6][7][8][9][10][11][12][13].…”