Anomalous diffusion and the first passage time problem

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 62, Issue 1 A, 2000, Pages 120-133

  Rangarajan, Govindan ^a,c   Ding, Mingzhou ^b  

^a INDIAN INSTITUTE OF SCIENCE   (India)

^b FLORIDA ATLANTIC UNIVERSITY   (United States)

^c JAWAHARLAL NEHRU CENTRE FOR ADVANCED SCIENTIFIC RESEARCH   (India)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFUSION; FOURIER TRANSFORMS; LAPLACE TRANSFORMS; NUMERICAL METHODS;

FIRST PASSAGE TIME (FPT); FOKKER-PLANCK EQUATION;

SURFACE CHEMISTRY;

EID: 0034227236     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.62.120     Document Type: Article

References (43)

1. H. Risken, The Fokker-Planck Equation (Springer-Verlag, Berlin, 1989).
2. C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1997).
3. H.C. Tuckwell, Introduction to Theoretical Neurobiology (Cambridge University Press, Cambridge, 1988), Vol. 2
4. Y. K. Lin and G. Q. Cai, Probabilistic Structural Dynamics (McGraw-Hill, New York, 1995).
5. M.F. Shlesinger, J. Klafter, and Y.M. Wong, J. Stat. Phys. 27, 499 (1982); J. Klafter, A. Blumen, and M.F. Shlesinger, Phys. Rev. A 35, 3081 (1987).
6. M.F. Shlesinger, J. Klafter, and Y.M. Wong, J. Stat. Phys. 27, 499 (1982); J. Klafter, A. Blumen, and M.F. Shlesinger, Phys. Rev. A 35, 3081 (1987).
7. A. Blumen, J. Klafter, and G. Zumofen, in Optical Spectroscopy of Glasses, edited by I. Zschokke (Reidel, Dodrecht, 1986); S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 (1987); J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 12 (1990); G. Zumofen, J. Klafter, and A. Blumen, J. Stat. Phys. 65, 991 (1991); Fractals in Biology and Medicine, edited by T. F. Nonnenmacher, G. A. Losa, and E. R. Weibl (Birkhauser, Basel, 1993); H.-P. Muller, R. Kimmich, and J. Weis, Phys. Rev. E 54, 5278 (1996); F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler, Phys. Rev. Lett. 77, 4470 (1996); A. Klemm, H.-P. Muller, and R. Kimmich, Phys. Rev. E 55, 4413 (1997); B.J. West, P. Grigolini, R. Metzler, and T.F. Nonnenmacher, ibid. 55, 99 (1997); Q. Gu, E. A. Schiff, S. Grebner, F. Wang, and R. Schwartz, Phys. Rev. Lett. 76, 3196 (1996); E.R. Weeks and H.L. Swinney, Phys. Rev. E 57, 4915 (1998).
8. A. Blumen, J. Klafter, and G. Zumofen, in Optical Spectroscopy of Glasses, edited by I. Zschokke (Reidel, Dodrecht, 1986); S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 (1987); J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 12 (1990); G. Zumofen, J. Klafter, and A. Blumen, J. Stat. Phys. 65, 991 (1991); Fractals in Biology and Medicine, edited by T. F. Nonnenmacher, G. A. Losa, and E. R. Weibl (Birkhauser, Basel, 1993); H.-P. Muller, R. Kimmich, and J. Weis, Phys. Rev. E 54, 5278 (1996); F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler, Phys. Rev. Lett. 77, 4470 (1996); A. Klemm, H.-P. Muller, and R. Kimmich, Phys. Rev. E 55, 4413 (1997); B.J. West, P. Grigolini, R. Metzler, and T.F. Nonnenmacher, ibid. 55, 99 (1997); Q. Gu, E. A. Schiff, S. Grebner, F. Wang, and R. Schwartz, Phys. Rev. Lett. 76, 3196 (1996); E.R. Weeks and H.L. Swinney, Phys. Rev. E 57, 4915 (1998).
9. A. Blumen, J. Klafter, and G. Zumofen, in Optical Spectroscopy of Glasses, edited by I. Zschokke (Reidel, Dodrecht, 1986); S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 (1987); J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 12 (1990); G. Zumofen, J. Klafter, and A. Blumen, J. Stat. Phys. 65, 991 (1991); Fractals in Biology and Medicine, edited by T. F. Nonnenmacher, G. A. Losa, and E. R. Weibl (Birkhauser, Basel, 1993); H.-P. Muller, R. Kimmich, and J. Weis, Phys. Rev. E 54, 5278 (1996); F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler, Phys. Rev. Lett. 77, 4470 (1996); A. Klemm, H.-P. Muller, and R. Kimmich, Phys. Rev. E 55, 4413 (1997); B.J. West, P. Grigolini, R. Metzler, and T.F. Nonnenmacher, ibid. 55, 99 (1997); Q. Gu, E. A. Schiff, S. Grebner, F. Wang, and R. Schwartz, Phys. Rev. Lett. 76, 3196 (1996); E.R. Weeks and H.L. Swinney, Phys. Rev. E 57, 4915 (1998).
10. A. Blumen, J. Klafter, and G. Zumofen, in Optical Spectroscopy of Glasses, edited by I. Zschokke (Reidel, Dodrecht, 1986); S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 (1987); J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 12 (1990); G. Zumofen, J. Klafter, and A. Blumen, J. Stat. Phys. 65, 991 (1991); Fractals in Biology and Medicine, edited by T. F. Nonnenmacher, G. A. Losa, and E. R. Weibl (Birkhauser, Basel, 1993); H.-P. Muller, R. Kimmich, and J. Weis, Phys. Rev. E 54, 5278 (1996); F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler, Phys. Rev. Lett. 77, 4470 (1996); A. Klemm, H.-P. Muller, and R. Kimmich, Phys. Rev. E 55, 4413 (1997); B.J. West, P. Grigolini, R. Metzler, and T.F. Nonnenmacher, ibid. 55, 99 (1997); Q. Gu, E. A. Schiff, S. Grebner, F. Wang, and R. Schwartz, Phys. Rev. Lett. 76, 3196 (1996); E.R. Weeks and H.L. Swinney, Phys. Rev. E 57, 4915 (1998).
11. A. Blumen, J. Klafter, and G. Zumofen, in Optical Spectroscopy of Glasses, edited by I. Zschokke (Reidel, Dodrecht, 1986); S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 (1987); J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 12 (1990); G. Zumofen, J. Klafter, and A. Blumen, J. Stat. Phys. 65, 991 (1991); Fractals in Biology and Medicine, edited by T. F. Nonnenmacher, G. A. Losa, and E. R. Weibl (Birkhauser, Basel, 1993); H.-P. Muller, R. Kimmich, and J. Weis, Phys. Rev. E 54, 5278 (1996); F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler, Phys. Rev. Lett. 77, 4470 (1996); A. Klemm, H.-P. Muller, and R. Kimmich, Phys. Rev. E 55, 4413 (1997); B.J. West, P. Grigolini, R. Metzler, and T.F. Nonnenmacher, ibid. 55, 99 (1997); Q. Gu, E. A. Schiff, S. Grebner, F. Wang, and R. Schwartz, Phys. Rev. Lett. 76, 3196 (1996); E.R. Weeks and H.L. Swinney, Phys. Rev. E 57, 4915 (1998).
12. A. Blumen, J. Klafter, and G. Zumofen, in Optical Spectroscopy of Glasses, edited by I. Zschokke (Reidel, Dodrecht, 1986); S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 (1987); J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 12 (1990); G. Zumofen, J. Klafter, and A. Blumen, J. Stat. Phys. 65, 991 (1991); Fractals in Biology and Medicine, edited by T. F. Nonnenmacher, G. A. Losa, and E. R. Weibl (Birkhauser, Basel, 1993); H.-P. Muller, R. Kimmich, and J. Weis, Phys. Rev. E 54, 5278 (1996); F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler, Phys. Rev. Lett. 77, 4470 (1996); A. Klemm, H.-P. Muller, and R. Kimmich, Phys. Rev. E 55, 4413 (1997); B.J. West, P. Grigolini, R. Metzler, and T.F. Nonnenmacher, ibid. 55, 99 (1997); Q. Gu, E. A. Schiff, S. Grebner, F. Wang, and R. Schwartz, Phys. Rev. Lett. 76, 3196 (1996); E.R. Weeks and H.L. Swinney, Phys. Rev. E 57, 4915 (1998).
13. A. Blumen, J. Klafter, and G. Zumofen, in Optical Spectroscopy of Glasses, edited by I. Zschokke (Reidel, Dodrecht, 1986); S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 (1987); J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 12 (1990); G. Zumofen, J. Klafter, and A. Blumen, J. Stat. Phys. 65, 991 (1991); Fractals in Biology and Medicine, edited by T. F. Nonnenmacher, G. A. Losa, and E. R. Weibl (Birkhauser, Basel, 1993); H.-P. Muller, R. Kimmich, and J. Weis, Phys. Rev. E 54, 5278 (1996); F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler, Phys. Rev. Lett. 77, 4470 (1996); A. Klemm, H.-P. Muller, and R. Kimmich, Phys. Rev. E 55, 4413 (1997); B.J. West, P. Grigolini, R. Metzler, and T.F. Nonnenmacher, ibid. 55, 99 (1997); Q. Gu, E. A. Schiff, S. Grebner, F. Wang, and R. Schwartz, Phys. Rev. Lett. 76, 3196 (1996); E.R. Weeks and H.L. Swinney, Phys. Rev. E 57, 4915 (1998).
14. A. Blumen, J. Klafter, and G. Zumofen, in Optical Spectroscopy of Glasses, edited by I. Zschokke (Reidel, Dodrecht, 1986); S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 (1987); J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 12 (1990); G. Zumofen, J. Klafter, and A. Blumen, J. Stat. Phys. 65, 991 (1991); Fractals in Biology and Medicine, edited by T. F. Nonnenmacher, G. A. Losa, and E. R. Weibl (Birkhauser, Basel, 1993); H.-P. Muller, R. Kimmich, and J. Weis, Phys. Rev. E 54, 5278 (1996); F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler, Phys. Rev. Lett. 77, 4470 (1996); A. Klemm, H.-P. Muller, and R. Kimmich, Phys. Rev. E 55, 4413 (1997); B.J. West, P. Grigolini, R. Metzler, and T.F. Nonnenmacher, ibid. 55, 99 (1997); Q. Gu, E. A. Schiff, S. Grebner, F. Wang, and R. Schwartz, Phys. Rev. Lett. 76, 3196 (1996); E.R. Weeks and H.L. Swinney, Phys. Rev. E 57, 4915 (1998).
15. A. Blumen, J. Klafter, and G. Zumofen, in Optical Spectroscopy of Glasses, edited by I. Zschokke (Reidel, Dodrecht, 1986); S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 (1987); J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 12 (1990); G. Zumofen, J. Klafter, and A. Blumen, J. Stat. Phys. 65, 991 (1991); Fractals in Biology and Medicine, edited by T. F. Nonnenmacher, G. A. Losa, and E. R. Weibl (Birkhauser, Basel, 1993); H.-P. Muller, R. Kimmich, and J. Weis, Phys. Rev. E 54, 5278 (1996); F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler, Phys. Rev. Lett. 77, 4470 (1996); A. Klemm, H.-P. Muller, and R. Kimmich, Phys. Rev. E 55, 4413 (1997); B.J. West, P. Grigolini, R. Metzler, and T.F. Nonnenmacher, ibid. 55, 99 (1997); Q. Gu, E. A. Schiff, S. Grebner, F. Wang, and R. Schwartz, Phys. Rev. Lett. 76, 3196 (1996); E.R. Weeks and H.L. Swinney, Phys. Rev. E 57, 4915 (1998).
16. A. Blumen, J. Klafter, and G. Zumofen, in Optical Spectroscopy of Glasses, edited by I. Zschokke (Reidel, Dodrecht, 1986); S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 (1987); J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 12 (1990); G. Zumofen, J. Klafter, and A. Blumen, J. Stat. Phys. 65, 991 (1991); Fractals in Biology and Medicine, edited by T. F. Nonnenmacher, G. A. Losa, and E. R. Weibl (Birkhauser, Basel, 1993); H.-P. Muller, R. Kimmich, and J. Weis, Phys. Rev. E 54, 5278 (1996); F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler, Phys. Rev. Lett. 77, 4470 (1996); A. Klemm, H.-P. Muller, and R. Kimmich, Phys. Rev. E 55, 4413 (1997); B.J. West, P. Grigolini, R. Metzler, and T.F. Nonnenmacher, ibid. 55, 99 (1997); Q. Gu, E. A. Schiff, S. Grebner, F. Wang, and R. Schwartz, Phys. Rev. Lett. 76, 3196 (1996); E.R. Weeks and H.L. Swinney, Phys. Rev. E 57, 4915 (1998).
17. A. Blumen, J. Klafter, and G. Zumofen, in Optical Spectroscopy of Glasses, edited by I. Zschokke (Reidel, Dodrecht, 1986); S. Havlin and D. Ben-Avraham, Adv. Phys. 36, 695 (1987); J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 12 (1990); G. Zumofen, J. Klafter, and A. Blumen, J. Stat. Phys. 65, 991 (1991); Fractals in Biology and Medicine, edited by T. F. Nonnenmacher, G. A. Losa, and E. R. Weibl (Birkhauser, Basel, 1993); H.-P. Muller, R. Kimmich, and J. Weis, Phys. Rev. E 54, 5278 (1996); F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler, Phys. Rev. Lett. 77, 4470 (1996); A. Klemm, H.-P. Muller, and R. Kimmich, Phys. Rev. E 55, 4413 (1997); B.J. West, P. Grigolini, R. Metzler, and T.F. Nonnenmacher, ibid. 55, 99 (1997); Q. Gu, E. A. Schiff, S. Grebner, F. Wang, and R. Schwartz, Phys. Rev. Lett. 76, 3196 (1996); E.R. Weeks and H.L. Swinney, Phys. Rev. E 57, 4915 (1998).
18. R. Metzler, J. Klafter, and I.M. Sokolov, Phys. Rev. E 58, 1621 (1998).
19. R. Metzler, E. Barkai, and J. Klafter, Phys. Rev. Lett. 82, 3563 (1999).
20. C. Fox, Trans. Am. Math. Soc. 98, 395 (1961).
21. A. M. Mathai and R. K. Saxena, The H-function with Applications in Statistics and Other Disciplines (Wiley Eastern, New Delhi, 1978).
22. X.-J. Wang, Phys. Rev. A 45, 8407 (1992).
23. G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes (Chapman and Hall, New York, 1994).
24. A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).
25. W. Feller, An Introduction to Probability Theory and Applications (Wiley, New York, 1971), Vol. 2.
26. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965).
27. M. Evans, N. Hastings, and B. Peacock, Statistical Distributions, 2nd ed. (Wiley, New York, 1993).
28. K. B. Oldham and J. Spanier, The Fractional Calculus (Academic, New York, 1974).
29. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993).
30. G. R. Grimmet and D. R. Stirzaker, Probability and Random Processes (Oxford University Press, New York, 1994).
31. H. M. Srivastava, K. C. Gupta, and S. P. Goyal, The H-functions of One and Two Variables with Applications (South Asian, New Delhi, 1982).
32. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
33. Higher Transcendental Functions, edited by A. Erdelyi (McGraw-Hill, New York, 1955), Vol. 3.
34. C. Fox, Proc. Am. Math. Soc. 29, 299 (1971).
35. G. Rangarajan and M. Ding (unpublished).
36. W. R. Schneider, in Stochastic Processes in Classical and Quantum Systems, edited by S. Albeverio, G. Casati, and D. Merlini (Springer, Berlin, 1986).
37. W.R. Schneider and W. Wyss, J. Math. Phys. 30, 134 (1989).
38. B.L.J. Braaksma, Compos. Math. 15, 239 (1964).
39. V. Balakrishnan, Physica A 132, 569 (1985).
40. S. Kullback, Information Theory and Statistics (Wiley, New York, 1968).
41. H. White, Estimation, Inference, and Specification Analysis (Cambridge University Press, New York, 1994).
42. D. R. Cox and H. D. Miller, Theory of Stochastic Processes (Methuen & Co., London, 1965).
43. A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (CRC Press, Boca Raton, FL, 1995).