Generalized (c,d)-Entropy and Aging Random Walks
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...
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2013-12-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/15/12/5324 |
| _version_ | 1817995903546425344 |
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| author | Rudolf Hanel Stefan Thurner |
| author_facet | Rudolf Hanel Stefan Thurner |
| author_sort | Rudolf Hanel |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-14T02:14:37Z |
| format | Article |
| id | doaj.art-958de9c6ee5843129fa4336c662950c9 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-14T02:14:37Z |
| publishDate | 2013-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-958de9c6ee5843129fa4336c662950c92022-12-22T02:18:17ZengMDPI AGEntropy1099-43002013-12-0115125324533710.3390/e15125324e15125324Generalized (c,d)-Entropy and Aging Random WalksRudolf Hanel0Stefan Thurner1Section for Science of Complex Systems, Medical University of Vienna, Spitalgasse 23, Vienna A-1090, AustriaSection for Science of Complex Systems, Medical University of Vienna, Spitalgasse 23, Vienna A-1090, AustriaComplex 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.http://www.mdpi.com/1099-4300/15/12/5324non-ergodicextensivitypath-dependencerandom walks with memory |
| spellingShingle | Rudolf Hanel Stefan Thurner Generalized (c,d)-Entropy and Aging Random Walks Entropy non-ergodic extensivity path-dependence random walks with memory |
| title | Generalized (c,d)-Entropy and Aging Random Walks |
| title_full | Generalized (c,d)-Entropy and Aging Random Walks |
| title_fullStr | Generalized (c,d)-Entropy and Aging Random Walks |
| title_full_unstemmed | Generalized (c,d)-Entropy and Aging Random Walks |
| title_short | Generalized (c,d)-Entropy and Aging Random Walks |
| title_sort | generalized c d entropy and aging random walks |
| topic | non-ergodic extensivity path-dependence random walks with memory |
| url | http://www.mdpi.com/1099-4300/15/12/5324 |
| work_keys_str_mv | AT rudolfhanel generalizedcdentropyandagingrandomwalks AT stefanthurner generalizedcdentropyandagingrandomwalks |