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Fractional Fokker-Planck equation for anomalous diffusion in a potential: Exact matrix continued fraction solutions

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Abstract

Methods for the exact solution of fractional Fokker-Planck equations for anomalous diffusion in an external potential are discussed using both ordinary and matrix continued fractions, whereby the scalar multi-term recurrence relations generated by such fractional diffusion equations are reduced to three-term matrix ones. The procedure is illustrated by solving various problems concerning the anomalous translational diffusion in both periodic and double-well potentials.

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References

  1. H. Risken, The Fokker-Planck Equation, 2nd edn. (Springer-Verlag, Berlin, 1989)

  2. R. Mazo, Brownian Motion: Fluctuations, Dynamics and Applications (Oxford University Press, Oxford, 2002)

  3. A. Einstein, in Investigations on the Theory of the Brownian Movement, edited by R.H. Fürth (Methuen, London, 1926)

  4. R. Hilfer, L. Anton, Phys. Rev. E 51, R848 (1995)

    Article  ADS  Google Scholar 

  5. R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. R. Metzler, J. Klafter, Adv. Chem. Phys. 116, 223 (2001)

    Article  Google Scholar 

  7. B.J. West, M. Bologna, P. Grigolini, Physics of Fractal Operators (Springer, New York, 2003)

  8. E.W. Montroll, M.F. Shlesinger, On the Wonderful World of Random Walks, edited by J.L. Lebowitz and E.W. Montroll, in Non-Equilibrium Phenomena II from Stochastics to Hydrodynamics (Elsevier Science Publishers, Amsterdam, 1984)

  9. E.W. Montroll, G.H. Weiss, J. Math. Phys. 6, 167 (1965)

    Article  MathSciNet  ADS  Google Scholar 

  10. W. Paul, J. Baschnagel, Stochastic Processes from Physics to Finance (Springer Verlag, Berlin, 1999)

  11. W.T. Coffey, Yu.P. Kalmykov, J.T. Waldron, Physica A 208, 462 (1994)

    Article  ADS  Google Scholar 

  12. W.T. Coffey, Yu.P. Kalmykov, The Langevin Equation, 3nd ed. (World Scientific, Singapore, 2012), Chapter 12

  13. H.S. Wall, Analytic Theory of Continued Fractions (Chelsea, New York, 1973)

  14. W.B. Jones, W.J. Thorn, Continued Fractions, Encyclopedia of Mathematics and its Applications, Vol. 11, edited by G.-C. Rota (Addison-Wesley, Reading, MA, USA, 1980)

  15. L. Lorentzen, H. Waadeland, Continued Fractions with Applications (North Holland, Amsterdam, 1992)

  16. A. Cuyt, V.B. Petersen, B. Verdonk, H. Waadel, W.B. Jones, Handbook of Continued Fractions for Special Functions (Springer, Berlin, 2008)

  17. H. Denk, M. Riederle, J. Approx. Theory 35, 355 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  18. W. Dieterich, P. Fulde, I. Peschel, Adv. Phys. 29, 527 (1980)

    Article  ADS  Google Scholar 

  19. P. Fulde, L. Pietronero, W.R. Schneider, S. Strässer, Phys. Rev. Lett. 35, 1776 (1975)

    Article  ADS  Google Scholar 

  20. C.J. Reid, Mol. Phys. 49, 331 (1983)

    Article  ADS  Google Scholar 

  21. F. Marchesoni, Phys. Rev. B 32, 1827 (1985)

    Article  ADS  Google Scholar 

  22. F. Marchesoni, J.K. Vij, Z. Phys. B 58, 187 (1985)

    Article  ADS  Google Scholar 

  23. B. Cartling, J. Chem. Phys. 90, 1819 (1989)

    Article  ADS  Google Scholar 

  24. H. Gang, A. Daffertshofer, H. Haken, Phys. Rev. Lett. 76, 4874 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. R. Ferrando, R. Spadacini, G.E. Tommei, Phys. Rev. E 48, 2437 (1993)

    Article  ADS  Google Scholar 

  26. W.T. Coffey, Yu.P. Kalmykov, S.V. Titov, B.P. Mulligan, Phys. Rev. E 73, 061101 (2006)

    Article  ADS  Google Scholar 

  27. W.L. Reenbohn, M.C. Mahato, J. Stat. Mech.: Theor. Exp., P03011 (2009)

  28. J.-D. Bao, Y. Zhou, K. Lü, Phys. Rev. E 74, 041125 (2006)

    Article  ADS  Google Scholar 

  29. E. Heinsalu, M. Patriarca, I. Goychuk, P. Hänggi, J. Phys.: Condens. Matter 19, 065114 (2007)

    Article  ADS  Google Scholar 

  30. Y.-M. Kang, J. Jiang, Y. Xie, J. Phys. A: Math. Theor. 44, 035002 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  31. Yu.P. Kalmykov, S.V. Titov, W.T. Coffey, Phys. Rev. E 85, 041101 (2012)

    Article  ADS  Google Scholar 

  32. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964)

  33. Yu.P. Kalmykov, W.T. Coffey, S.V. Titov, Phys. Rev. E, 74, 011105 (2006)

    Article  ADS  Google Scholar 

  34. F. So, K.L. Liu, Physica A 331, 378 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  35. C.W. Chow, K.L. Liu, Physica A 341, 87 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  36. Y.M. Kang, Y.L. Jiang, Phys. Rev. E 81, 021109 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  37. W.T. Coffey, Yu.P. Kalmykov, S.V. Titov, Adv. Chem. Phys. 133B, 285 (2006)

    Article  Google Scholar 

  38. R. Metzler, Phys. Rev. E 62, 6233 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  39. R. Metzler, J. Klafter, J. Phys. Chem. B 104, 3851 (2000)

    Article  Google Scholar 

  40. E. Barkai, R.S. Silbey, J. Phys. Chem. B 104, 3866 (2000)

    Article  Google Scholar 

  41. W.T. Coffey, Yu.P. Kalmykov, S.V. Titov, Phys. Rev. E 67, 061115 (2003)

    Article  ADS  Google Scholar 

  42. J. Voit, The Statistical Mechanics of Financial Markets, 2nd edn. (Springer-Verlag, Berlin, 2003)

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Coffey, W.T., Kalmykov, Y.P. & Titov, S.V. Fractional Fokker-Planck equation for anomalous diffusion in a potential: Exact matrix continued fraction solutions. Eur. Phys. J. Spec. Top. 222, 1847–1856 (2013). https://doi.org/10.1140/epjst/e2013-01968-x

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  • DOI: https://doi.org/10.1140/epjst/e2013-01968-x

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