Journal of Statistical Physics

, Volume 146, Issue 4, pp 762–773

Noise-Induced Drift in Stochastic Differential Equations with Arbitrary Friction and Diffusion in the Smoluchowski-Kramers Limit

Article

Abstract

We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e.g. Brownian motion. We study the limit where friction effects dominate the inertia, i.e. where the mass goes to zero (Smoluchowski-Kramers limit). Using the Itô stochastic integral convention, we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation, which can be parametrized by α∈ℝ. Interestingly, in addition to the classical Itô (α=0), Stratonovich (α=0.5) and anti-Itô (α=1) integrals, we show that position-dependent α=α(x), and even stochastic integrals with α∉[0,1] arise. Our findings are supported by numerical simulations.

Keywords

Brownian motion Stochastic differential equations Smoluchowski-Kramers approximation Einstein mobility-diffusion relation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Øksendal, B.: Stochastic Differential Equations. Springer, Berlin (1998) Google Scholar
  2. 2.
    Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. Springer, New York (1998) Google Scholar
  3. 3.
    Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1981) MATHGoogle Scholar
  4. 4.
    Lyons, T.J., Caruana, M., Lévy, T.: Differential equations driven by rough paths. Lecture Notes in Mathematics, vol. 1908. Springer, Berlin (2007). Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, With an introduction concerning the Summer School by Jean Picard MATHGoogle Scholar
  5. 5.
    Turelli, M.: Theor. Popul. Biol. 12, 140 (1977) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Ao, P.: Commun. Theor. Phys. 49, 1073 (2008) MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Gardiner, C.: Handbook of Stochastic Methods. Springer, Berlin (1985) Google Scholar
  8. 8.
    Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1999) Google Scholar
  9. 9.
    Sussmann, H.: Ann. Probab., 19–41 (1978) Google Scholar
  10. 10.
    Ermark, D.L., McCammon, J.A.: J. Chem. Phys. 69, 1352 (1978) ADSCrossRefGoogle Scholar
  11. 11.
    Lançon, P., Batrouni, G., Lobry, L., Ostrowsky, N.: EPL (Europhys. Lett.) 54, 28 (2001) ADSCrossRefGoogle Scholar
  12. 12.
    Lau, A.W.C., Lubensky, T.C.: Phys. Rev. E 76, 011123 (2007) MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Volpe, G., Helden, L., Brettschneider, T., Wehr, J., Bechinger, C.: Phys. Rev. Lett. 104, 170602 (2010) ADSCrossRefGoogle Scholar
  14. 14.
    Brettschneider, T., Volpe, G., Helden, L., Wehr, J., Bechinger, C.: Phys. Rev. E 83, 041113 (2011) ADSCrossRefGoogle Scholar
  15. 15.
    Wehr, J.: University of Arizona (2011, in preparation) Google Scholar
  16. 16.
    Ao, P., Kwon, C., Qian, H.: Complexity 12, 19 (2007) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Papanicolaou: In: AMS Seminar, Lecture Notes. Rensselaer Polytechnic Institute, Troy (1975). Google Scholar
  18. 18.
    Pavliotis, G.A., Stuart, A.M.: Multiscale Methods: Averaging and Homogenization. Springer, Berlin (2008) MATHGoogle Scholar
  19. 19.
    Schuss, Z.: Theory and Application of Stochastic Differential Equations. Wiley, New York (1980) Google Scholar
  20. 20.
    Pardoux, È., Veretennikov, A.: Ann. Probab. 29 (2001) Google Scholar
  21. 21.
    Pardoux, È., Veretennikov, A.: Ann. Probab. 31 (2003) Google Scholar
  22. 22.
    Pardoux, È., Veretennikov, A.: Ann. Probab. 33 (2005) Google Scholar
  23. 23.
    Smoluchowski, M.: Phys. Z. 17, 557 (1916) ADSGoogle Scholar
  24. 24.
    Kramers, H.: Physica 7, 284 (1940) MathSciNetADSMATHCrossRefGoogle Scholar
  25. 25.
    Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967) MATHGoogle Scholar
  26. 26.
    Freidlin, M.: J. Stat. Phys. 117, 617 (2004) MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    Kupferman, R., Pavliotis, G.A., Stuart, A.M.: Phys. Rev. E 70, 036120 (2004) MathSciNetADSCrossRefGoogle Scholar
  28. 28.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics. Interscience, New York (1953) Google Scholar
  29. 29.
    Lax, P.: Functional Analysis. Wiley, New York (2002) MATHGoogle Scholar
  30. 30.
    Pavliotis, G., Stuart, A.: Multiscale Model. Simul. 4, 1 (2005) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Sigurgeirsson, H., Stuart, A.M.: Phys. Fluids 14, 4352 (2002) MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Program in Applied MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Max-Planck-Institut für Intelligente SystemeStuttgartGermany
  3. 3.2. Physikalisches InstitutUniversität StuttgartStuttgartGermany
  4. 4.Department of PhysicsBilkent UniversityAnkaraTurkey

Personalised recommendations