{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T17:27:05Z","timestamp":1770571625188,"version":"3.49.0"},"reference-count":48,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2012,4,10]],"date-time":"2012-04-10T00:00:00Z","timestamp":1334016000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper we utilize the Tsallis relative entropy, a generalization of the Kullback\u2013Leibler entropy in the frame work of non-extensive thermodynamics to analyze the properties of anomalous diffusion processes. Anomalous (super-) diffusive behavior can be described by fractional diffusion equations, where the second order space derivative is extended to fractional order \u03b1 \u2208 (1, 2). They represent a bridging regime, where for \u03b1 = 2 one obtains the diffusion equation and for \u03b1 = 1 the (half) wave equation is given. These fractional diffusion equations are solved by so-called stable distributions, which exhibit heavy tails and skewness. In contrast to the Shannon or Tsallis entropy of these distributions, the Kullback and Tsallis relative entropy, relative to the pure diffusion case, induce a natural ordering of the stable distributions consistent with the ordering implied by the pure diffusion and wave limits.<\/jats:p>","DOI":"10.3390\/e14040701","type":"journal-article","created":{"date-parts":[[2012,4,10]],"date-time":"2012-04-10T11:09:50Z","timestamp":1334056190000},"page":"701-716","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":39,"title":["Tsallis Relative Entropy and Anomalous Diffusion"],"prefix":"10.3390","volume":"14","author":[{"given":"Janett","family":"Prehl","sequence":"first","affiliation":[{"name":"Institute of Physics, Chemnitz University of Technology, D-09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christopher","family":"Essex","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, University of Western Ontario, Middlesex College, London, ON, N6A 5B7, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Karl Heinz","family":"Hoffmann","sequence":"additional","affiliation":[{"name":"Institute of Physics, Chemnitz University of Technology, D-09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,4,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1080\/00018738700101072","article-title":"Diffusion in disordered media","volume":"36","author":"Havlin","year":"1987","journal-title":"Adv. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1209\/0295-5075\/13\/6\/009","article-title":"Anomalous diffusion of hydrogen in amorphous metals","volume":"13","author":"Schirmacher","year":"1990","journal-title":"Europhys. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3518","DOI":"10.1529\/biophysj.104.044263","article-title":"Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells","volume":"87","author":"Weiss","year":"2004","journal-title":"Biophys. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2960","DOI":"10.1529\/biophysj.104.051078","article-title":"Anomalous diffusion of proteins due to molecular crowding","volume":"89","author":"Banks","year":"2005","journal-title":"Biophys. J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2950","DOI":"10.1016\/S0006-3495(96)79865-X","article-title":"Anomalous diffusion of water in biological tissues","volume":"70","author":"Corinth","year":"1996","journal-title":"Biophys. J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/S0301-0104(02)00546-3","article-title":"Subdiffusion-limited reactions","volume":"284","author":"Yuste","year":"2002","journal-title":"Chem. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"198101","DOI":"10.1103\/PhysRevLett.94.198101","article-title":"Optimal search strategies for hidden targets","volume":"94","author":"Coppey","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"020102","DOI":"10.1103\/PhysRevE.74.020102","article-title":"Two-dimensional intermittent search processes: An alternative to L\u00e9vy flight strategies","volume":"74","author":"Loverdo","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1038\/443281a","article-title":"Mathematical physics: Search research","volume":"443","author":"Shlesinger","year":"2006","journal-title":"Nature"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3975","DOI":"10.1103\/PhysRevLett.71.3975","article-title":"Observation of anomalous diffusion and L\u00e9vy flights in a two-dimensional rotating flow","volume":"71","author":"Solomon","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Pekalski, A., and Kutner, R. (1999). Anomalous Diffusion from Basics to Application, Springer-Verlag. Lecture Notes in Physics, 519.","DOI":"10.1007\/BFb0106828"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1063\/1.528578","article-title":"Fractional diffusion and wave equation","volume":"30","author":"Schneider","year":"1989","journal-title":"J. Math. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/0378-4371(92)90441-R","article-title":"Fractional diffusion equation for transport phenomena in random media","volume":"185","author":"Giona","year":"1992","journal-title":"Physica A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/0378-4371(94)90064-7","article-title":"Fractional model equation for anomalous diffusion","volume":"211","author":"Metzler","year":"1994","journal-title":"Physica A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","article-title":"The random walk\u2019s guide to anomalous diffusion: A fractional dynamics approach","volume":"339","author":"Metzler","year":"2000","journal-title":"Phys. Rep."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific Publishing.","DOI":"10.1142\/9789812817747"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Klages, R., Radons, G., and Sokolov, I.M. (2008). Anomalous Transport: Foundations and Applications, Wiley-VCH. Chapter 6.","DOI":"10.1002\/9783527622979"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"5501","DOI":"10.1088\/0305-4470\/33\/31\/305","article-title":"The similarity group and anomalous diffusion equations","volume":"33","author":"Schulzky","year":"2000","journal-title":"J. Phys. A: Math. Gen."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"11441","DOI":"10.1088\/1751-8113\/40\/38\/001","article-title":"Modeling anomalous superdiffusion","volume":"40","author":"Fischer","year":"2007","journal-title":"J. Phys. A: Math. Gen."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Klages, R., Radons, G., and Sokolov, I.M. (2008). Anomalous Transport\u2014Foundations and Applications, Wiley-VCH. [1st ed.].","DOI":"10.1002\/9783527622979"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"030101(R)","DOI":"10.1103\/PhysRevE.63.030101","article-title":"Nonlinear equation for anomalous diffusion: Unified power-law and stretched exponential exact solution","volume":"63","author":"Malacarne","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"031106","DOI":"10.1103\/PhysRevE.72.031106","article-title":"Logarithmic diffusion and porous media equations: a unified description","volume":"72","author":"Pedron","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1515\/JNETDY.2003.017","article-title":"Fractional diffusion, irreversibility and entropy","volume":"28","author":"Li","year":"2003","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1016\/j.physa.2009.09.009","article-title":"The superdiffusion entropy production paradox in the space-fractional case for extended entropies","volume":"389","author":"Prehl","year":"2010","journal-title":"Physica A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1515\/jnet.1998.23.2.166","article-title":"Fractional diffusion and entropy production","volume":"23","author":"Hoffmann","year":"1998","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/S0378-4371(00)00174-6","article-title":"Tsallis and R\u00e9nyi entropies in fractional diffusion and entropy production","volume":"284","author":"Essex","year":"2000","journal-title":"Physica A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On information and sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Schl\u00f6gl, F. (1989). Probability and Heat, Vieweg.","DOI":"10.1007\/978-3-663-13977-5"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1109\/24.722289","article-title":"Testing exponentiality of the residual life, based on dynamic Kullback\u2013Leibler information","volume":"47","author":"Ebrahimi","year":"1998","journal-title":"IEEE Trans. Reliab."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1287\/opre.1070.0385","article-title":"Relative entropy, exponential utility, and robust dynamic pricing","volume":"55","author":"Lim","year":"2007","journal-title":"Oper. Res."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Kekre, H.B., Bharadi, V.A., Shaktia, P., Shah, V., and Ambardekar, A.A. (2011, January 25\u201326). Keystroke dynamic analysis using relative entropy & timing sequence Euclidean distance. Proceedings of the International Conference and Workshop on Emerging Trends in Technology (ICWET 2011), Mumbai, India.","DOI":"10.1145\/1980022.1980072"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Yu, S., and Mehta, P.G. (2009, January 16\u201318). The Kullback\u2013Leibler rate metric for comparing dynamical systems. Proceedings of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, China.","DOI":"10.1109\/CDC.2009.5399552"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1109\/LSP.2003.809034","article-title":"Fast approximation of Kullback\u2013Leibler distance for dependence trees and hidden Markov models","volume":"10","author":"Do","year":"2003","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/0003-4916(88)90094-2","article-title":"Complexity as thermodynamic depth","volume":"188","author":"Llody","year":"1988","journal-title":"Ann. Physics"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1103\/PhysRevE.59.275","article-title":"Thermodynamic depth of casual states: Objective complexity via minimal representation","volume":"59","author":"Crutchfield","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1103\/RevModPhys.74.197","article-title":"The role of relative entropy in quantum information theory","volume":"74","author":"Vedral","year":"2002","journal-title":"Rev. Mod. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2910","DOI":"10.1109\/TIT.2003.819324","article-title":"Kullback\u2013Leibler approximation of spectral density functions","volume":"49","author":"Georgiou","year":"2003","journal-title":"IEEE Trans. Inform. Theor."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/0375-9601(96)00365-9","article-title":"Geometry of quantum inference","volume":"219","author":"Braunstein","year":"1996","journal-title":"Phys. Lett. A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"080501\u20131\u20134","DOI":"10.1103\/PhysRevLett.104.080501","article-title":"Unified view of quantum and classical correlations","volume":"104","author":"Modi","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1109\/83.982822","article-title":"Wavelet-based texture retrieval using generalized Gaussian density and Kullback\u2013Leibler distance","volume":"11","author":"Do","year":"2002","journal-title":"IEEE Trans. Image Process."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1071\/WR99107","article-title":"Kullback\u2013Leibler information as a basis for strong inference in ecological studies","volume":"28","author":"Burnham","year":"2001","journal-title":"Wildl. Res."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1442","DOI":"10.1103\/PhysRevE.58.1442","article-title":"Generalized entropy-based criterion for consistent testing","volume":"58","author":"Tsallis","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Borland\n\t\t\t\t\t\t\t              L.\n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t              Plastino\n\t\t\t\t\t\t\t              A.R.\n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t              Tsallis\n\t\t\t\t\t\t\t              C.\n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t          Information gain within nonextensive thermostatics\n\t\t\t\t\t\t\t          J. Math. Phys.\n\t\t\t\t\t\t\t          1998\n\t\t\t\t\t\t\t          39\n\t\t\t\t\t\t\t          6490\n\t\t\t\t\t\t\t          6501\n\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t              Borland\n\t\t\t\t\t\t\t              L.\n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t              Plastino\n\t\t\t\t\t\t\t              A.R.\n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t              Tsallis\n\t\t\t\t\t\t\t              C.\n\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t          Erratum: \u201cInformation gain with generalized termostatistics\u201d[J. Math. Phys. 39, 6490 (1998)]\n\t\t\t\t\t\t\t          J. Math. Phys.\n\t\t\t\t\t\t\t          1999\n\t\t\t\t\t\t\t          40\n\t\t\t\t\t\t\t          2196\n\t\t\t\t\t\t\t          10.1063\/1.532660","DOI":"10.1063\/1.533119"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"4868","DOI":"10.1063\/1.1805729","article-title":"Fundamental properties of Tsallis relative entropy","volume":"45","author":"Furuichi","year":"2004","journal-title":"J. Math. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"3927","DOI":"10.1103\/PhysRevE.56.3927","article-title":"Minimum Kullback entropy approach to the Fokker\u2013Planck equation","volume":"56","author":"Plastino","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_46","unstructured":"Nolan, J.P. Chapter 3: Personal communication."},{"key":"ref_47","unstructured":"Samorodnitsky, G., and Taqqu, M.S. (1994). Stable non-Gaussian Random Processes, Chapman & Hall."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"043303","DOI":"10.1063\/1.3104063","article-title":"Escort mean values and the characterization of power-law-decaying probability densities","volume":"50","author":"Tsallis","year":"2009","journal-title":"J. Math. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/4\/701\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:49:43Z","timestamp":1760219383000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/4\/701"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,4,10]]},"references-count":48,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2012,4]]}},"alternative-id":["e14040701"],"URL":"https:\/\/doi.org\/10.3390\/e14040701","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,4,10]]}}}