- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Fractional dynamics from the ordinary Langevin equation

A. A. Stanislavsky*

  • Institute of Radio Astronomy, 4 Chervonopraporna St., Kharkov 61002, Ukraine

  • *Electronic address: alexstan@ira.kharkov.ua
PDF

Phys. Rev. E 67, 021111 – Published 27 February, 2003

DOI: https://doi.org/10.1103/PhysRevE.67.021111

Abstract

We consider the usual Langevin equation depending on an internal time. This parameter is substituted by a first passage time of a self-similar Markov process. Then the Gaussian process is parent, and the hitting time process is directing. The probability to find the resulting process at the real time is defined by the integral relationship between the probability densities of the parent and directing processes. The corresponding master equation becomes the fractional Fokker-Planck equation. We show that the resulting process has non-Markovian properties, all its moments are finite, the fluctuation-dissipation relation and the H-theorem hold.

View figure in article

References (31)

  1. W.T. Coffey, Yu.P. Kalmykov, and J.T. Waldron, The Langevin Equation with Applications in Physics, Chemistry and Electrical Engineering (World Scientific, Singapore, 1996).
  2. N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1984).
  3. K. Hasselmann, Tellus 28, 473 (1976).
  4. L. Arnold, Random Dynamical Systems (Springer-Verlag, Berlin, 1998).
  5. H. Hasegawa and T. Nakagomi, J. Stat. Phys. 21, 191 (1979).
  6. H.C. Tuckwell, Stochastic Processes in the Neurosciences (SIAM, Philadelphia, 1989).
  7. J.P. Bouchaud and A. Georges, Phys. Rep. 195, 127 (1990).
  8. R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).
  9. H. Mori, Prog. Theor. Phys. 33, 423 (1965).
  10. R. Kubo, Rep. Prog. Phys. 29, 255 (1966).
  11. M. Płoszajczak and T. Srokowski, Ann. Phys. (N.Y.) 249, 236 (1996).
  12. R. Muralidhar, D. Ramkrishna, H. Nakanishi, and D.J. Jacobs, Physica A 167, 539 (1990).
  13. K. Lindenberg and B.J. West, The Nonequilibrium Statistical Mechanics of Open and Closed Systems (VCH, New York, 1990).
  14. N.H. Bingham, Z. Wahrscheinlichkeitstheor. Verwandte Geb. 17, 1 (1971).
  15. R. Höpfner, Scand. J. Stat. 17, 201 (1990).
  16. W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1971).
  17. H.C. Fogedby, Phys. Rev. E 50, 1657 (1994); ibid. 58, 1690 (1998).
  18. I.M. Sokolov, Phys. Rev. E 63, 056111 (2001).
  19. S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives—Theory and Applications (Gordon and Breach, New York, 1993).
  20. R. Metzler, Phys. Rev. E 62, 6233 (2000); R. Metzler and J. Klafter, J. Phys. Chem. B 104, 3851 (2000).
  21. E. Barkai, R. Metzler, and J. Klafter, Phys. Rev. E 61, 132 (2000).
  22. A.A. Stanislavsky, Phys. Rev. E 61, 4752 (2000).
  23. R. Metzler, J. Klafter, and I.M. Sokolov, Phys. Rev. E 58, 1621 (1998).
  24. A.I. Saichev and G.M. Zaslavsky, Chaos 7, 753 (1997).
  25. R. Metzler, E. Barkai, and J. Klafter, Phys. Rev. Lett. 82, 3563 (1999).
  26. G. Rangarajan and M. Ding, Fractals 8, 139 (2000).
  27. E. Barkai and R.J. Silbey, J. Phys. Chem. B 104, 3866 (2000).
  28. E.W. Montroll and G.H. Weiss, J. Math. Phys. 6, 167 (1965).
  29. M.M. Meerschaert and H. Scheffler, Limit Theorems for Continuous Time Random Walks), http://unr.edu/homepage/mcubed/LimitCTRW.pdf
  30. I.M. Sokolov, J. Klafter, and A. Blumen, Phys. Rev. E 64, 021107 (2001).
  31. W.G. Glöckle and T.F. Nonnenmacher, Macromolecules 24, 6426 (1991).

Outline

Information

Phys. Rev. E 67, 021111– Published 27 February, 2003
  • Received 15 August 2002

© 2003 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation