Abstract
The nonadditive entropy Sq 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 Sq 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.
Highlights
In the 1870s Ludwig Boltzmann introduced a microscopic expression for the thermodynamic entropy introduced by Clausius a few years earlier in the frame of thermodynamics
For reasons that will soon become clear, we shall here refer to this theory as the Boltzmann–Gibbs (BG) statistical mechanics
One may say that the developments in this area along the last two decades strongly suggest that the epistemological process that is occurring exhibits some similarity with what happened at the beginning of the 1900s with Mechanics, nowadays frequently referred to as Newtonian mechanics, known to be particular limits of both relativistic mechanics and quantum mechanics
Summary
In the 1870s Ludwig Boltzmann introduced a microscopic expression for the thermodynamic entropy introduced by Clausius a few years earlier in the frame of thermodynamics. This Gaussian form is known to be consistent with the Maxwellian distribution of velocities for classical statistical mechanics, with the solution of the standard Fokker–Planck equation in the presence of the most general quadratic potential, and with the classical Central Limit Theorem (CLT) The latter basically states that if we consider the sum SN = N i=1 Xi of N independent (or nearly independent in some sense) random variables {Xi }, each of them having a finite variance, this sum converges for N → ∞, after appropriate centering and rescaling, to a Gaussian. No general rigorous first-principle proof (i.e., just using mechanics and theory of probabilities, with no other hypothesis) yet exists for classical (or quantum) Hamiltonian systems [3], there remains—after 140 years of impressive success—no reasonable doubt that the BG entropy is the correct one to be used for a wide and important class of physical systems, basically those whose (nonlinear) dynamics is strongly chaotic (meaning, for classical systems, positive maximal Lyapunov exponent), mixing, ergodic. This q-generalization and its applications are briefly reviewed in the rest of the present paper, which is based on various previous books and reviews [7,8,9,10,11,12], parts of which are here followed/reproduced for simplicity and self-completeness
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