Approach of complexity in nature: Entropic nonuniqueness

Abstract

Boltzmann introduced in the 1870's a logarithmic measure for the connection between the thermodynamical entropy and the probabilities of the microscopic configurations of the system. His entropic functional for classical systems was extended by Gibbs to the entire phase space of a many-body system, and by von Neumann in order to cover quantum systems as well. Finally, it was used by Shannon within the theory of information. The simplest expression of this functional corresponds to a discrete set of $W$ microscopic possibilities, and is given by $S_{BG}=-k{\sum }_{i=1}^W{p}_i\ln p_i$ ($k$ is a positive universal constant; {\it BG} stands for {\it Boltzmann-Gibbs}). This relation enables the construction of BG statistical mechanics. The BG theory has provided uncountable important applications. Its application in physical systems is legitimate whenever the hypothesis of {\it ergodicity} is satisfied. However, {\it what can we do when ergodicity and similar simple hypotheses are violated?}, which indeed happens in very many natural, artificial and social complex systems. It was advanced in 1988 the possibility of generalizing BG statistical mechanics through a family of nonadditive entropies, namely $S_q=k\frac{1-{\sum }_{i=1}^W{p}_i^q}{q-1}$, which recovers the additive $S_{BG}$ entropy in the $q\to 1$ limit. The index $q$ is to be determined from mechanical first principles. Along three decades, this idea intensively evolved world-wide (see Bibliography in \url{http://tsallis.cat.cbpf.br/biblio.htm}), and led to a plethora of predictions, verifications, and applications in physical systems and elsewhere. As expected whenever a {\it paradigm shift} is explored, some controversy naturally emerged as well in the community. The present status of the general picture is here described, starting from its dynamical and thermodynamical foundations, and ending with its most recent physical applications.

15 pages including 3 figures. To appear in Axioms. Guest Editor: Prof. Dr. Hans J. Haubold. The present invited overview belongs to the Special Issue "Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. Arak M. Mathai at the occasion of his 80th Birthday"

References in corpus (17)

• The large deviation approach to statistical mechanics
• Pathway Model, Superstatistics, Tsallis Statistics, and a Generalized Measure of Entropy
• From QCD-based hard-scattering to nonextensive statistical mechanical descriptions of transverse momentum spectra in high-energy $pp$ and $p\bar{p}$ collisions
• Description of High-Energy pp Collisions Using Tsallis Thermodynamics: Transverse Momentum and Rapidity Distributions
• Zeroth Law compatibility of non-additive thermodynamics
• Numerical indications of a q-generalised central limit theorem
• Generalized Measure of Entropy, Mathai's Distributional Pathway Model, and Tsallis Statistics
• Tsallis Distribution Decorated With Log-Periodic Oscillation
• Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle
• On generalized entropy measures and pathways
• On a Generalized Entropy Measure Leading to the Pathway Model: with a preliminary application to solar neutrino data
• Possible Implication of a Single Nonextensive $p_T$ Distribution for Hadron Production in High-Energy $pp$ Collisions
• System size dependence of the log-periodic oscillations of transverse momentum spectra
• Comment on "Towards a large deviation theory for strongly correlated systems"
• Boltzmann-Gibbs entropy is sufficient but not necessary for the likelihood factorization required by Einstein
• Pathway parameter and thermonuclear functions
• Emergence of $q$-statistical functions in a generalized binomial distribution with strong correlations

Cited by in corpus (1)

• Analytic approaches of the anomalous diffusion: a review