Non-additive entropies and statistical mechanics at the edge of chaos: a bridge between natural and social sciences

Abstract

The Boltzmann–Gibbs (BG) statistical mechanics constitutes one of the pillars of contemporary theoretical physics. It is constructed upon the other pillars—classical, quantum, relativistic mechanics and Maxwell equations for electromagnetism—and its foundations are grounded on the optimization of the BG (additive) entropic functional S BG = − k ∑ i p i ln ⁡ p i . Its use in the realm of classical mechanics is legitimate for vast classes of nonlinear dynamical systems under the assumption that the maximal Lyapunov exponent is positive (currently referred to as strong chaos ), and its validity has been experimentally verified in countless situations. It fails however when the maximal Lyapunov exponent vanishes (referred to as weak chaos ), which is virtually always the case with complex natural, artificial and social systems. To overcome this type of weakness of the BG theory, a generalization was proposed in 1988 grounded on the non-additive entropic functional S q = k ( ( 1 − ∑ i p i q ) / ( q − 1 ) ) ( q ∈ R ; S 1 = S BG ) . The index q and related ones are to be calculated, whenever mathematically tractable, from first principles and reflect the specific class of weak chaos. We review here the basics of this generalization and illustrate its validity with selected examples aiming to bridge natural and social sciences. This article is part of the theme issue ‘Thermodynamics 2.0: Bridging the natural and social sciences (Part 2)’.

The Multiscale Principle in Nature (Principium luxuriæ): Linking Multiscale Thermodynamics to Living and Non-Living Complex Systems

“…The lack of clarity regarding the multiscale influence of one system on another leads to the idea that very different disciplines (such as physics, life sciences, and humanities, among others) are in separate fields and their unification is seen as unthinkable or improbable [271,272]. However, new developments in the field of thermodynamics have questioned this separation, so recent efforts seek to unify the sciences under the lens of thermodynamics, stability, and dissipation [273][274][275][276][277][278][279][280][281][282][283]. The way in which excess external energy is dissipated at different scales can be seen as a description that aligns with the perspective of unifying different scientific fields through thermodynamics as parts of a "big history".…”

Assessing structural integrity of non-homogeneous systems by means of Acoustic Emissions and Non-Extensive Statistical Mechanics

“…Considering δτ as the X parameter of Eqn. (5), it has been proven experimentally [3,16,18,19] that the CDF of the IT intervals obeys a q-exponential function into the form of:…”

“…A series of non-additive entropies have been introduced, however, the one that is nowadays most widely adopted is the S q entropy, which was proposed about thirty-five years ago by Tsallis [1]. S q has been employed successfully for the analysis of problems in a broad variety of disciplines either in laboratory [2,3] or in-field scales (ranging from Strength of Materials, Seismology, and Earthquake Engineering [4] to Finance, Medicine and Social Sciences [5]). The concepts of NESM were recently applied in the direction of detecting pre-failure indices while brittle building materials (like marble and concrete) are submitted to mechanical loading schemes, including direct tension, three-point bending, compression, and diametral compression etc.…”