Abstract

The pillars of contemporary theoretical physics are classical mechanics, Maxwell electromagnetism, relativity, quantum mechanics, and Boltzmann–Gibbs (BG) statistical mechanics –including its connection with thermodynamics. The BG theory describes amazingly well the thermal equilibrium of a plethora of so-called simple systems. However, BG statistical mechanics and its basic additive entropy started, in recent decades, to exhibit failures or inadequacies in an increasing number of complex systems. The emergence of such intriguing features became apparent in quantum systems as well, such as black holes and other area-law-like scenarios for the von Neumann entropy. In a different arena, the efficiency of the Shannon entropy—as the BG functional is currently called in engineering and communication theory—started to be perceived as not necessarily optimal in the processing of images (e.g., medical ones) and time series (e.g., economic ones). Such is the case in the presence of generic long-range space correlations, long memory, sub-exponential sensitivity to the initial conditions (hence vanishing largest Lyapunov exponents), and similar features. Finally, we witnessed, during the last two decades, an explosion of asymptotically scale-free complex networks. This wide range of important systems eventually gave support, since 1988, to the generalization of the BG theory. Nonadditive entropies generalizing the BG one and their consequences have been introduced and intensively studied worldwide. The present review focuses on these concepts and their predictions, verifications, and applications in physics and elsewhere. Some selected examples (in quantum information, high- and low-energy physics, low-dimensional nonlinear dynamical systems, earthquakes, turbulence, long-range interacting systems, and scale-free networks) illustrate successful applications. The grounding thermodynamical framework is briefly described as well.

Highlights

  • Relativity generalizes Newtonian mechanics in order to include velocities close to that of light; along a different line, quantum mechanics generalizes Newtonian mechanics in order to include small masses

  • We have argued that the additivity of the by Gibbs (BG) entropy accommodates extremely well, as is extensively known, to those many systems whose elements are generically not very strongly correlated and/or whose nonlinear dynamics are governed by strong chaos, meaning that their dynamics are associated to a sensitivity to initial conditions that exponentially diverge with time

  • It fails for those complex systems that do not satisfy such requirements, violating, in particular, the thermodynamic extensivity expected for their entropy and whose dynamical sensitivity to initial conditions diverges sub-exponentially with time

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Summary

Introduction

Relativity generalizes Newtonian mechanics in order to include velocities close to that of light; along a different line, quantum mechanics generalizes Newtonian mechanics in order to include small masses. The entropy SBG and its associated statistical mechanics enable the correct calculation of a large variety of thermostatistical properties at or near the thermal equilibrium of uncountable so-called simple systems When it comes to wide classes of so-called complex systems, the BG theory fails. We focus on some selected illustrations, based on the volume of applications and connections that are exhibited in the constantly growing literature

Rényi Entropy
Kaniadakis Entropy and κ-Exponential Distribution
Beck–Cohen Superstatistics
More Entropies and Applications
Thermodynamical Background
Conclusions and Perspectives
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