Next Article in Journal
Physical Properties of High Entropy Alloys
Next Article in Special Issue
Structural Patterns in Complex Systems Using Multidendrograms
Previous Article in Journal
New Results on Fractional Power Series: Theories and Applications
Previous Article in Special Issue
Generalized Statistical Mechanics at the Onset of Chaos
Article Menu

Export Article

Open AccessArticle
Entropy 2013, 15(12), 5324-5337;

Generalized (c,d)-Entropy and Aging Random Walks

1,* and 1,2,3,*
Section for Science of Complex Systems, Medical University of Vienna, Spitalgasse 23, Vienna A-1090, Austria
Santa Fe Institute,1399 Hyde Park Road, Santa Fe, NM87501, USA
Institute for Applied Systems Analysis, Schlossplatz 1, Laxenburg A-2361, Austria
Authors to whom correspondence should be addressed.
Received: 26 September 2013 / Revised: 12 November 2013 / Accepted: 25 November 2013 / Published: 3 December 2013
(This article belongs to the Special Issue Complex Systems)
View Full-Text   |   Download PDF [474 KB, uploaded 24 February 2015]   |  


Complex systems are often inherently non-ergodic and non-Markovian and Shannon entropy loses its applicability. Accelerating, path-dependent and aging random walks offer an intuitive picture for non-ergodic and non-Markovian systems. It was shown that the entropy of non-ergodic systems can still be derived from three of the Shannon–Khinchin axioms and by violating the fourth, the so-called composition axiom. The corresponding entropy is of the form Sc,d ~ ∑iΓ(1 + d, 1 − cln pi) and depends on two system-specific scaling exponents, c and d. This entropy contains many recently proposed entropy functionals as special cases, including Shannon and Tsallis entropy. It was shown that this entropy is relevant for a special class of non-Markovian random walks. In this work, we generalize these walks to a much wider class of stochastic systems that can be characterized as “aging” walks. These are systems whose transition rates between states are path- and time-dependent. We show that for particular aging walks, Sc,d is again the correct extensive entropy. Before the central part of the paper, we review the concept of (c,d)-entropy in a self-contained way. View Full-Text
Keywords: non-ergodic; extensivity; path-dependence; random walks with memory non-ergodic; extensivity; path-dependence; random walks with memory

Figure 1

This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

Share & Cite This Article

MDPI and ACS Style

Hanel, R.; Thurner, S. Generalized (c,d)-Entropy and Aging Random Walks. Entropy 2013, 15, 5324-5337.

Show more citation formats Show less citations formats

Related Articles

Article Metrics

Article Access Statistics



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