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Entropy 2013, 15(12), 5324-5337; doi:10.3390/e15125324

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

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)
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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.
Keywords: non-ergodic; extensivity; path-dependence; random walks with memory non-ergodic; extensivity; path-dependence; random walks with memory
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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

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