Time evolution of relative entropies for anomalous diffusion

Entropy

Volumn 15, Issue 8, 2013, Pages 2989-3006

  Prehl, Janett ^a   Boldt, Frank ^a   Essex, Christopher ^b   Hoffmann, Karl Heinz ^a  

^a CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

^b UNIVERSITY OF WESTERN ONTARIO   (Canada)

Author keywords

Kullback Leibler entropy; Space fractional diffusion equation; Stable distribution; Tsallis relative entropy

Indexed keywords

EID: 84884194766     PISSN: None     EISSN: 10994300     Source Type: Journal    
DOI: 10.3390/e15082989     Document Type: Article

References (46)

1. Kullback, S.; Leibler, R.A. On information and sufficiency. Ann. Math. Stat. 1951, 22, 79-86.
2. Schlögl, F. Probability and Heat; Vieweg: Braunschweig, Germany, 1989.
3. Tsallis, C. Generalized entropy-based criterion for consistent testing. Phys. Rev. E 1998, 58, 1442-1445.
4. Borland, L.; Plastino, A.R.; Tsallis, C. Information gain within nonextensive thermostatics. J. Math. Phys. 1998, 39, 6490-6501.
5. Borland, L.; Plastino, A.R.; Tsallis, C. Erratum: "Information gain within generalized thermostatistics" [J. Math. Phys. 39, 6490 (1998)]. J. Math. Phys. 1999, 40, 2196.
6. Furuichi, S.; Yanagi, K.; Kuriyama, K. Fundamental properties of Tsallis relative entropy. J. Math. Phys. 2004, 45, 4868-4877.
7. Yamano, T. A possible extension of Shannon's information theory. Entropy 2001, 3, 280-292.
8. Yu, S.; Mehta, P.G. The Kullback-Leibler Rate Metric for Comparing Dynamical Systems. In Proceedings of IEEE Conference on Decision and Control, joint with 28th Chinese Control Conference, Shanghai, China, 16-18 December 2009; pp. 8363-8368.
9. Anastasiadis, A. Special issue: Tsallis entropy. Entropy 2012, 14, 174-176.
10. Zhang, Y.; Wu, L. Optimal multi-level thresholding based on maximum Tsallis entropy via an artificial bee colony approach. Entropy 2011, 13, 841-859.
11. Nelson, K.P.; Scannell, B.J.; Landau, H. A risk profile for information fusion algorithms. Entropy 2011, 13, 1518-1532.
12. Balasis, G.; Daglis, I.A.; Papadimitriou, C.; Anastasiadis, A.; Sandberg, I.; Eftaxias, K. Quantifying dynamical complexity of magnetic storms and solar flares via nonextensive Tsallis entropy. Entropy 2011, 13, 1865-1881.
13. Telesca, L. Tsallis-based nonextensive aAnalysis of the Southern California seismicity. Entropy 2011, 13, 1267-1280.
14. Ribeiro, M.S.; Nobre, F.D.; Curado, E.M.F. Classes of n-dimensional nonlinear Fokker-Planck equations associated to Tsallis entropy. Entropy 2011, 13, 1928-1944.
15. Weiss, M.; Elsner, M.; Kartberg, F.; Nilsson, T. Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells. Biophys. J. 2004, 87, 3518-3524.
16. Banks, D.S.; Fradin, C. Anomalous diffusion of proteins due to molecular crowding. Biophys. J. 2005, 89, 2960-2971.
17. Yuste, S.B.; Lindberg, K. Subdiffusion-limited reactions. Chem. Phys. 2002, 284, 169-180.
18. Havlin, S.; Ben-Avraham, D. Diffusion in disordered media. Adv. Phys. 1987, 36, 695-798.
19. Schirmacher, W.; Perm, M.; Suck, J.B.; Heidemann, A. Anomalous diffusion of hydrogen in amorphous metals. Europhys. Lett. 1990, 13, 523-529.
20. Solomon, T.R.;Weeks, E.R.; Swinney, H.L. Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow. Phys. Rev. Lett. 1993, 71, 3975-3978.
21. Hansen, A.E.; Jullien, M.C.; Paret, J.; Tabeling, P. Dispersion in Freely Decaying and Forced 2D Turbulence. In Anomalous Diffusion: From Basics to Applications; Pekalski, A., Kutner, R., Eds.; Lecture notes in physics; Springer-Verlag: Berlin, Germany, 1999; pp. 186-196.
22. Bénichou, O.; Coppey, M.; Moreau, M.; Suet, P.H.; Voituriez, R. Optimal search strategies for hidden targets. Phys. Rev. Lett. 2005, 94, 198101.
23. Bénichou, O.; Loverdo, C. Moreau, M.; Voituriez, R. Two-dimensional intermittent search processes: An alternative to Lévy flight strategies. Phys. Rev. E 2006, 74, 020102.
24. Shlesinger, M.F. Mathematical physics: Search research. Nature 2006, 443, 281-282.
25. Barthelemy, P.; Bertolotti, J.;Wiersma, D.S. A Lévy flight for light. Nature 2008, 453, 495-498.
26. Malacarne, L.C.; Mendes, R.S.; Pedron, I.T.; Lenzi, E.K. Nonlinear equation for anomalous diffusion: Unified power-law and stretched exponential exact solution. Phys. Rev. E 2001, 63, 030101.
27. Pedron, I.T.; Mendes, R.S.; Buratta, T.J.; Malacarne, L.C.; Lenzi, E.K. Logarithmic diffusion and porous media equations: A unified description. Phys. Rev. E 2005, 72, 031106.
28. Schneider, W.R.; Wyss, W. Fractional diffusion and wave equation. J. Math. Phys. 1989, 30, 134-144.
29. Giona, M.; Roman, H.E. Fractional diffusion equation for transport phenomena in random media. Physica A 1992, 185, 87-97.
30. Metzler, R.; Glöckle, W.G.; Nonnenmacher, T.F. Fractional model equation for anomalous diffusion. Physica A 1994, 211, 13-24.
31. Metzler, R.; Klafter, J. The random walk's guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 2000, 339, 1-77.
32. Hilfer, R. Applications of Fractional Calculus in Physics; World Scientific Publishing: Singapore, Singapore, 2000.
33. del Castillo-Negrete, D. Fractional Diffusional Models of Anomalous Transport. In Anomalous Transport: Foundations and Applications, 1st ed.; Klages, R., Radons, G., Sokolov, I.M., Eds.; Wiley-VCH: Weinheim, Germany, 2008; chapter 6, pp. 163-212.
34. Schulzky, C.; Essex, C.; Davison, M.; Franz, A.; Hoffmann, K.H. The similarity group and anomalous diffusion equations. J. Phys. A: Math. Gen. 2000, 33, 5501-5511.
35. Fischer, A.; Seeger, S.; Hoffmann, K.H.; Essex, C.; Davison, M. Modeling anomalous superdiffusion. J. Phys. A: Math. Gen. 2007, 40, 11441-11452.
36. Hoffmann, K.H.; Prehl, J. Anomalous Transport in Disordered Fractals. In Anomalous Transport: Foundations and Applications, 1st ed.; Klages, R., Radons, G., Sokolov, I.M., Eds.; Wiley-VCH: Weinheim, Germany, 2008.
37. Li, X.; Essex, C.; Davison, M.; Hoffmann, K.H.; Schulzky, C. Fractional diffusion, irreversibility and entropy. J. Non-Equilib. Thermodyn. 2003, 28, 279-291.
38. Dybiec, B.; Gudowska-Nowak, E.; Hänggi, P. Lévy-Brownian motion on finite intervals: Mean first passage time analysis. Phys. Rev. E 2006, 73, 046104.
39. Prehl, J.; Essex, C.; Hoffmann, K.H. The superdiffusion entropy production paradox in the space-fractional case for extended entropies. Physica A 2010, 389, 215-224.
40. Hoffmann, K.H.; Essex, C.; Schulzky, C. Fractional diffusion and entropy production. J. Non-Equilib. Thermodyn. 1998, 23, 166-175.
41. Essex, C.; Schulzky, C.; Franz, A.; Hoffmann, K.H. Tsallis and Rényi entropies in fractional diffusion and entropy production. Physica A 2000, 284, 299-308.
42. Hoffmann, K.H.; Essex, C.; Prehl, J. A unified approach to resolving the entropy production paradox. J. Non-Equilib. Thermodyn. 2012, 37, 393-412.
43. Prehl, J.; Essex, C.; Hoffmann, K.H. Tsallis relative entropy and anomalous diffusion. Entropy 2012, 14, 701-716.
44. Nolan, J.P. Stable Distributions-Models for Heavy Tailed Data; Birkhäuser: Boston, MA, USA, 2012. in progress; Chapter 1, Available online: http://academic2.american.edu/jpnolan/stable/stable.html/ (accessed on 30 March 2012); Chapter 3, private communication.
45. Bender, C.M.; Orszag, S.A. Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory; McGraw Hill: New York, NY, USA, 1978.
46. Daniels, H.E. Saddleppoint approximation in statistics. Ann. Math. Stat. 1954, 25, 631-650.