Exploring the Neighborhood of q-Exponentials.

Henrique Santos Lima,Constantino Tsallis

doi:10.3390/e22121402

Henrique Santos Lima, Constantino Tsallis

Open Access

https://doi.org/10.3390/e22121402

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Abstract

The q-exponential form is obtained by optimizing the nonadditive entropy (with , where BG stands for Boltzmann–Gibbs) under simple constraints, and emerges in wide classes of natural, artificial and social complex systems. However, in experiments, observations and numerical calculations, it rarely appears in its pure mathematical form. It appears instead exhibiting crossovers to, or mixed with, other similar forms. We first discuss departures from q-exponentials within crossover statistics, or by linearly combining them, or by linearly combining the corresponding q-entropies. Then, we discuss departures originated by double-index nonadditive entropies containing as particular case.

Highlights

It turns out that wide classes of complex systems can be satisfactorily handled within a generalization of Boltzmann–Gibbs (BG) statistical mechanics based on the nonadditive entropy q
We focus on crossover statistics (Section 2), linear combinations of q-exponential functions (Section 3), linear combinations of q-entropies (Section 4), BR [15] and S 0 [16] (Section 5)
Crossover statistics is often useful whenever the phenomenon which is focused on exhibits a q-exponential behavior within a range of the relevant variables, and makes a crossover to another q-exponential function with a different index q

Summary

Introduction

Nonadditive entropies have been used as a basis to explain a diversity of phenomena, from astrophysics to the oscillatory behavior of El Niño [1,2,3], from DNA to financial markets [4,5]. It turns out that wide classes of complex systems can be satisfactorily handled within a generalization of Boltzmann–Gibbs (BG) statistical mechanics based on the nonadditive entropy q. The aim of the present article is to discuss in detail some departures from a pure q-exponential function which frequently emerge in real situations. Such variations are used in the statistics of nucleotides in full genomes [4], the re-association of folded proteins [12], standard map for intermediate values of the control parameter [13], to mention but a few. We focus on crossover statistics (Section 2), linear combinations of q-exponential functions (Section 3), linear combinations of q-entropies (Section 4), BR [15] and S 0 [16] (Section 5)

Multiple Crossover Statistics

Linear Combination of Normalized q-Exponentials

Linear Combination of q-Entropies

Other Departures—Two-Indices Entropies

Conclusions