Inter-occurrence times and universal laws in finance, earthquakes and genomes

doi:10.1016/j.chaos.2015.12.025

Chaos, Solitons & Fractals

Volume 88, July 2016, Pages 254-266

https://doi.org/10.1016/j.chaos.2015.12.025 Get rights and content

Abstract

A plethora of natural, artificial and social systems exist which do not belong to the Boltzmann–Gibbs (BG) statistical-mechanical world, based on the standard additive entropy S_BG and its associated exponential BG factor. Frequent behaviors in such complex systems have been shown to be closely related to q-statistics instead, based on the nonadditive entropy S_q (with $S_{1} = S_{B G}$ ), and its associated q-exponential factor which generalizes the usual BG one. In fact, a wide range of phenomena of quite different nature exist which can be described and, in the simplest cases, understood through analytic (and explicit) functions and probability distributions which exhibit some universal features. Universality classes are concomitantly observed which can be characterized through indices such as q. We will exhibit here some such cases, namely concerning the distribution of inter-occurrence (or inter-event) times in the areas of finance, earthquakes and genomes.

Section snippets

Historical and physical motivations

In 1865 Clausius introduced in thermodynamics, and named, the concept of entropy (noted S, probably in honor of Sadi Carnot, whom Clausius admired) [1]. It was introduced on completely macroscopic terms, with no reference at all to the microscopic world, whose existence was under strong debate at his time, and still even so several decades later. One of the central properties of this concept was to be thermodynamically extensive, i.e., to be proportional to the size of the system (characterized

Thermodynamical entropic extensivity in strongly correlated systems generically mandates nonadditive entropic functionals

In what follows we shall refer to uncorrelated or weakly correlated N-body systems whenever Eq. (7) occurs, and to strongly correlated ones whenever zero-Lebesgue-measure behaviors such as those in Eqs. (8) and (9) occur.

Let us introduce now the following entropic functional ( $q \in R$ ): $S_{q} = k \frac{1 - \sum_{i = 1}^{W} p_{i}^{q}}{q - 1} (S_{1} = S_{B G}) .$ This expression can be equivalently rewritten as follows: $\begin{matrix} S_{q} = k \sum_{i = 1}^{W} p_{i} \ln_{q} \frac{1}{p_{i}} = - k \sum_{i = 1}^{W} p_{i}^{q} \ln_{q} p_{i} = - k \sum_{i = 1}^{W} p_{i} \ln_{2 - q} p_{i}, \end{matrix}$ where $\ln_{q} z \equiv \frac{z^{1 - q} - 1}{1 - q} (\ln_{1} z = \ln z) .$ For the particular instance of equal probabilities

Why should the thermodynamical entropy always be extensive?

In what follows we focus on arguments yielding, as final outcome, that the thermodynamical entropy of any system must be extensive. These arguments follow along three different lines, namely thermodynamical mathematical structure, large deviation theory, and time evolution of the entropy of nonlinear dynamical systems towards their stationary states.

Inter-occurrence times in finance, earthquakes and genomes

Let us now focus on an interesting universal property, shared by many natural, artificial and social systems constituted by strongly-correlated elements, namely the distribution of specific inter-occurrence times. We will specially review some available results in finance [130], [131], earthquakes [132], and biology [133], among others [80], [81], [134]. We will not include in the present occasion the discussion of long-range-interacting Hamiltonian classical systems [82], [135], [136], [137],

Final remarks

We have discussed here a variety of phenomena of the type that typically exists in financial theory, but which also emerge in many other complex systems such as earthquakes and genetics. More precisely, the distribution of inter-occurrence times (or distances) appears again and again to be of the q-exponential form. This universal law, though with values for (q, β) that depend on the specific universality class that we are focusing on, plays a simple and relevant role. The calculation of the

Acknowledgments

F.C. Alcaraz, C. Beck, E.P. Borges, T. Bountis, A. Bunde, L.J.L. Cirto, A. Coniglio, E.M.F. Curado, M. Gell-Mann, P. Grigolini, G. Ruiz, H.J. Herrmann, M. Jauregui, H.J. Jensen, J. Ludescher, F.D. Nobre, A. Pluchino, S.M.D. Queiros, A. Rapisarda, P. Rapcan, P. Tempesta, U. Tirnakli, S. Umarov, and many others, are warmly acknowledged for old and recent fruitful discussions. I also acknowledge partial financial support by the Brazilian agencies CNPq and FAPERJ, and by the John Templeton

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^☆: Invited review to appear in Chaos, Solitons and Fractals.

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Inter-occurrence times and universal laws in finance, earthquakes and genomes☆

Abstract

Section snippets

Historical and physical motivations

Thermodynamical entropic extensivity in strongly correlated systems generically mandates nonadditive entropic functionals

Why should the thermodynamical entropy always be extensive?

Inter-occurrence times in finance, earthquakes and genomes

Final remarks

Acknowledgments

EPL

Phys Lett A

Phys Lett A

Phys Rev E

Phys Lett A

Phys Rev D

Nucl Phys B

Phys Rev Lett

Phys Rev D

J Mol Liq

Eur Phys J B

JSTAT

Eur Phys J B

J Math Phys

Physica A

Phys Rev E

Phys Rev E

EPJ

Chemical Physics

Phys Rev E

Physica D

über verschiedene für die anwendung bequeme formen der hauptgleichungen der mechanischen wärmetheorie

Ann Phys

Weitere studien über das wärmegleichgewicht unter gas molekülen [Further studies on thermal equilibrium between gas molecules]

Wien Berl

über die beziehung eines allgemeine mechanischen satzes zum zweiten haupsatze der wärmetheorie

Sitzungsberichte, K. Akademie der Wissenschaften in Wien Math Naturwissenschaften

Elementary principles in statistical mechanics – developed with especial reference to the rational foundation of thermodynamics

J Stat Phys

Proc Natl Acad Sci USA

Introduction to nonextensive statistical mechanics – approaching a complex world

Physica A

Eur Phys J A

Ann Phys

Phys Lett A

Eur Phys J C

Phys Rev E

Phys Rev E

Europhys Lett

Proc Natl Acad Sci USA

Phys Rev E

Proc R Soc A

Phys Rev Lett

Phys Rev E

Phys Lett A

J Phys A Math Gen

J Phys A Math Gen

EPL

J Stat Mech Theory Exp

Nucl Phys B

Physica A

Thermodynamics and an introduction to thermostatistics

Phys Rev B

Phys Lett A

Phys Rev B

Phys Rev B