Next Article in Journal
Entropy in Urban Systems
Next Article in Special Issue
Generalized (c,d)-Entropy and Aging Random Walks
Previous Article in Journal
Non–Parametric Estimation of Mutual Information through the Entropy of the Linkage
Previous Article in Special Issue
Co-Evolutionary Mechanisms of Emotional Bursts in Online Social Dynamics and Networks
Entropy 2013, 15(12), 5178-5222; doi:10.3390/e15125178

Generalized Statistical Mechanics at the Onset of Chaos

Instituto de Física y Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, Circuito de la Investigación Científica Ciudad Universitaria, Ciudad de Mexico 04510, Mexico
Received: 4 October 2013 / Revised: 9 November 2013 / Accepted: 11 November 2013 / Published: 27 November 2013
(This article belongs to the Special Issue Complex Systems)
Download PDF [2468 KB, uploaded 24 February 2015]


Transitions to chaos in archetypal low-dimensional nonlinear maps offer real and precise model systems in which to assess proposed generalizations of statistical mechanics. The known association of chaotic dynamics with the structure of Boltzmann–Gibbs (BG) statistical mechanics has suggested the potential verification of these generalizations at the onset of chaos, when the only Lyapunov exponent vanishes and ergodic and mixing properties cease to hold. There are three well-known routes to chaos in these deterministic dissipative systems, period-doubling, quasi-periodicity and intermittency, which provide the setting in which to explore the limit of validity of the standard BG structure. It has been shown that there is a rich and intricate behavior for both the dynamics within and towards the attractors at the onset of chaos and that these two kinds of properties are linked via generalized statistical-mechanical expressions. Amongst the topics presented are: (i) permanently growing sensitivity fluctuations and their infinite family of generalized Pesin identities; (ii) the emergence of statistical-mechanical structures in the dynamics along the routes to chaos; (iii) dynamical hierarchies with modular organization; and (iv) limit distributions of sums of deterministic variables. The occurrence of generalized entropy properties in condensed-matter physical systems is illustrated by considering critical fluctuations, localization transition and glass formation. We complete our presentation with the description of the manifestations of the dynamics at the transitions to chaos in various kinds of complex systems, such as, frequency and size rank distributions and complex network images of time series. We discuss the results.
Keywords: low dimensional chaos; statistical physics; complex systems low dimensional chaos; statistical physics; complex systems
This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Share & Cite This Article

Further Mendeley | CiteULike
Export to BibTeX |
MDPI and ACS Style

Robledo, A. Generalized Statistical Mechanics at the Onset of Chaos. Entropy 2013, 15, 5178-5222.

View more citation formats

Related Articles

Article Metrics

For more information on the journal, click here


Cited By

[Return to top]
Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert