Nonadditive entropy and nonextensive statistical mechanics -an overview after 20 years

Abstract

Statistical mechanics constitutes one of the pillars of contemporary physics. Recognized as such -together with mechanics (classical, quantum, relativistic), electromagnetism and thermodynamics -, it is one of the mandatory theories studied at virtually all the intermediate-and advanced-level courses of physics around the world. As it normally happens with such basic scientific paradigms, it is placed at a crossroads of various other branches of knowledge. In the case of statistical mechanics, the standard theory -hereafter referred to as the Boltzmann-Gibbs (BG) statistical mechanics -exhibits highly relevant connections at a variety of microscopic, mesoscopic and macroscopic physical levels, as well as with the theory of probabilities (in particular, with the Central Limit Theorem, CLT ). In many circumstances, the ubiquitous efects of the CLT , with its Gaussian attractors (in the space of the distributions of probabilities), are present. Within this complex ongoing frame, a possible generalization of the BG theory was advanced in 1988 (C.T., J. Stat. Phys. 52, 479). The extension of the standard concepts is intended to be useful in those "pathological", and nevertheless very frequent, cases where the basic assumptions (molecular chaos hypothesis, ergodicity) for applicability of the BG theory would be violated. Such appears to be, for instance, the case in classical long-range-interacting many-body Hamiltonian systems (at the so-called quasi-stationary state). Indeed, in such systems, the maximal Lyapunov exponent vanishes in the thermodynamic limit N → ∞. This fact creates a quite novel situation with regard to typical BG systems, which generically have a positive value for this central nonlinear dynamical quantity. This peculiarity has sensible effects at all physical micro-, meso-and macroscopic levels. It even poses deep challenges at the level of the CLT . In the present occasion, after 20 years of the 1988 proposal, we undertake here an overview of some selected successes of the approach, and of some interesting points that still remain as open questions. Various theoretical, experimental, observational and computational aspects will be addressed.

Investigation of an energy nonadditivity for nonextensive systems

“… characterizes the energy nonadditivity of the system under consideration and should be constant for a given system in a given situation with constant E(AB). The same logic can be found in [4] for a similar nonadditivity in entropy with many examples of empirically determined  for entropy.…”

Nonextensive Electrical and Thermostatistical Parametric Dependency on Temperature and Tsallis Factor by the Formulation of Q-Exponential Integrals

“…As, we know that the Lyapunov exponent diminishes in a non-extensive domain as the thermodynamic limit approaches to infinity [1][2][3] in an ergodic system [1][2][3]. The molecular chaos theory [1][2][3][4] is applicable to this case because a meniscus amount of scattering occurs due to the defects present in the lattice at the lower temperature values. Weak chaos of the molecules does occur along with the long range interactions of the colder ions which are generated due to a temperature residing across to the absolute zero.…”

“…Nonextensive statistical mechanics is one of the contemporary pillars of physics which was formulated in the year 1988 in the famous Journal of Statistical Physics known to be as "Possible generalization of the Boltzmann-Gibbs statistics" [1]. This new era of statistical mechanics is being used to classify several kinds of entropy which exists in the system, to calculate it and to find the possibility of its reduction inside the system to some extent.…”

“…This branch extends the empire of Statistical Mechanics while validating its applicability and its interconnectivity between metaphysics, mesoscopic physics and macroscopic physics. This branch is based upon the theory of probabilities i.e., CLT (Central Limit Theorem) [1]. Based upon the CLT the Fermi-Dirac distribution function with an introduction of the Tsallis factor q [1][2][3] is given by ( ) ( ) ( ) The effective electron concentration has been calculated by the integral of the product of the distribution function along with the density of states of the electron gas forming a q-exponential integral.…”

Generalized entropy indices to measure α- and β-diversities of macrophytes

“…The Patil and Taillie's indices have been intensely studied recently in the context of statistical physics [25][26][27] and are usually written in the form:…”