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

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

1,*  and 1,2,3,*
Received: 26 September 2013; in revised form: 12 November 2013 / Accepted: 25 November 2013 / Published: 3 December 2013
(This article belongs to the Special Issue Complex Systems)
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Abstract: 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 which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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MDPI and ACS Style

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

AMA Style

Hanel R, Thurner S. Generalized (c,d)-Entropy and Aging Random Walks. Entropy. 2013; 15(12):5324-5337.

Chicago/Turabian Style

Hanel, Rudolf; Thurner, Stefan. 2013. "Generalized (c,d)-Entropy and Aging Random Walks." Entropy 15, no. 12: 5324-5337.

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