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Fractional Stochastics

  • Bruce J. West
  • Mauro Bologna
  • Paolo Grigolini
Chapter
  • 411 Downloads
Part of the Institute for Nonlinear Science book series (INLS)

Abstract

The modeling of complex phenomena using random walks was discussed earlier. In that discussion we outlined how a simple random walk model of a normal diffusion process leads to Gaussian statistics and a mean-square displacement that increases linearly with time. We also examined how inverse power-law memory in the random fluctuations, that is, in the steps of the walker, can produce a system response that is anomalous in that the mean-square displacement is proportional to t 2H , where 0 < H < 1. We saw that such time series are random fractals with fractal dimension given by D = 2 - H. The most complex phenomena studied earlier involved the limit of fractional differences becoming fractional derivatives, so that a stochastic process with long-term memory can be generated by taking the fractional derivative of a Wiener process. We learned that such processes have Gaussian statistics, but they also have inverse power-law spectra.

Keywords

Master Equation Fractional Derivative Langevin Equation Fractional Brownian Motion Wait Time Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. [1]
    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, US Dept. of Commerce, NBS, Appl. Math. Ser. 55 (1972).Google Scholar
  2. [2]
    P. S. Addison, IAHR J. Hydraulic Research 34, 5439 (1996).Google Scholar
  3. [3]
    P. S. Addison and A. S. Ndumu, Engineering applications of fractional Brownian motion: self-affine and self-similar random processes, Fractals 7, 151 (1999).CrossRefGoogle Scholar
  4. [4]
    P. Allegrini, M. Barbi, P. Grigolini and B. J. West, Dynamical Model for DNA sequences, Phys. Rev. E 52, 5281–96 (19995).CrossRefGoogle Scholar
  5. [5]
    M. Bologna, P. Grigolini and J. Rioccardi, The Lévy diffusion as an effect of sporadic randomness, submitted to Phys. Rev. E Google Scholar
  6. [6]
    G. Cattaneo, Atti. Sem. Mat. Fis. Univ. Modena 3, 83 (1948).MathSciNetGoogle Scholar
  7. [7]
    N. Chakravaxti and K. L. Sevastian, Chem. Phys. Lett. 267, 9 (1997).ADSCrossRefGoogle Scholar
  8. [8]
    E. L. Chen, P. C. Chung, H. M. Tsai and C. I. Cheng, IEEE Trans. Biomed. Eng. 45, 783 (1998).CrossRefGoogle Scholar
  9. [9]
    W. N. Findley, J. S. Lai and K. Onaran, Creep and Relaxation of Nonlinear Viscoelastic Materials, Dover, New York (1976).zbMATHGoogle Scholar
  10. [10]
    T. Geisel and S. Thomas, Phys. Rev. Lett. 52, 1936 (1984).MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    W. G. Glöckle and T. F. Nonnenmacher, Fractional relaxation and the time-temperature superposition principle, Rheol. Acta 33 (1994) 337.CrossRefGoogle Scholar
  12. [12]
    D. G. Le Grand, W. V. Olszewski and J. T. Bendler, J. Pol. Sci. BB 25, 1149 (1987);CrossRefGoogle Scholar
  13. [12A]
    J. T. Bendler and M. F. Shlesinger, J. Stat. Phys. 53, 531 (1988).ADSCrossRefGoogle Scholar
  14. [13]
    I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged edition, Academic, New York (1980).zbMATHGoogle Scholar
  15. [14]
    A. M. Hammad and M. A. Issa, Adv. Cement Based Mat. 1, 169 (1994).CrossRefGoogle Scholar
  16. [15]
    M. Jaroniec, Reac. Kinet. Catal Lett. 8, 425 (1978)CrossRefGoogle Scholar
  17. [16]
    G. Jumarie, Stochastic differential equations with fractional Brownian motion inputs, Int. J. Systems Sci. 24, 1113 (1993).MathSciNetzbMATHCrossRefGoogle Scholar
  18. [17]
    V. Kobelev and E. Romanov, Fractional Langevin Equation to describe anomalous diffusion, Prog. Theor. Phys. Supp. 139, 470–476 (2000).ADSCrossRefGoogle Scholar
  19. [18]
    K. M. Kolwankar and A. D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos 6, 505 (1996).MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. [19]
    B. B. Mandelbrot and J. W. van Ness, Fractional Brownian motions, fractional noise and applications, SIAM Rev. 10, 422 (1968).MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. [20]
    K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, New York (1993).zbMATHGoogle Scholar
  22. [21]
    R. R. Nigmatullin, Theor. and Math. Phys. 90(3), 245 (1992).MathSciNetADSCrossRefGoogle Scholar
  23. [22]
    A. Nordseick, W. E. Lamb and G. E. Uhlenbeck, Physica 7, 344 (1940).MathSciNetADSCrossRefGoogle Scholar
  24. [23]
    D. J. Odde, E. M. Tanaka, S. S. Hawkins and J. M. Buettner, Biotech. and Bioeng. 50, 452 (1996).CrossRefGoogle Scholar
  25. [24]
    I. Oppenheim, K. Shuler and G. Weiss, The Master Equation, MIT University Press, Cambridge, MA (1977).Google Scholar
  26. [25]
    W. Pauli, Festschrift zum60 gebürtstag A. Sommerfeld, S. Hirzel, Leipzig (1928).Google Scholar
  27. [26]
    A. Rocco and B. J. West, Fractional calculus and the evolution of fractal phenomena, Physica A 265, 535 (1999).CrossRefGoogle Scholar
  28. [27]
    M. F. Shlesinger, B. J. West and J. Klafter, Lévy dynamics for enhanced diffusion: an application to turbulence, Phys. Rev. Lett. 58, 1100–03 (1987).MathSciNetADSCrossRefGoogle Scholar
  29. [28]
    A. A. Stanislavsky, Memory effects and macroscopic manifestation of randomness, Phys. Rev. E 61, 4752–4759 (2000).ADSCrossRefGoogle Scholar
  30. [29]
    G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion, Phys. Rev. 36, 823 (1930).ADSCrossRefGoogle Scholar
  31. [30]
    G. Tréfan, P. Grigolmi and B. J. West, Deterministic Brownian Motion, Phys. Rev. A 45, 1249 (1992).ADSCrossRefGoogle Scholar
  32. [31]
    M. O. Vlad, An inverse scaling approach to multi-state random activation energy model, Physica A 184, 303–324 (1992).MathSciNetADSCrossRefGoogle Scholar
  33. [32]
    M. O. Vlad, J. Coll. Interface Sci. 128, 388 (1989).CrossRefGoogle Scholar
  34. [33]
    B. J. West and V. Seshadri, Linear systems with Lévy fluctuations, Physica A 113, 203–216 (1982).MathSciNetADSCrossRefGoogle Scholar
  35. [34]
    B. J. West and M. F. Shlesinger,Random walk of dislocations following a high velocity impact, J. Stat. Phys. 30, 527 (1983);ADSCrossRefGoogle Scholar
  36. [34A]
    B. J. West and M. F. Shlesinger Random walk model of impact phenomena, Physica 127 A, 490 (1984).zbMATHCrossRefGoogle Scholar
  37. [35]
    A. van der Ziel, Physica 10, 359 (1950).CrossRefGoogle Scholar
  38. [36]
    R. W. Zwanzig, Physica 30, 1109 (1964).MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 2003

Authors and Affiliations

  • Bruce J. West
    • 1
  • Mauro Bologna
    • 2
  • Paolo Grigolini
    • 2
  1. 1.Department of the Army, U.S. Army Research LaboratoryArmy Research OfficeResearch Triangle ParkUSA
  2. 2.Department of PhysicsUniversity of North TexasDentonUSA

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