Journal of Statistical Physics

, Volume 146, Issue 4, pp 762–773 | Cite as

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

  • Scott HottovyEmail author
  • Giovanni Volpe
  • Jan Wehr


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.


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


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  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) zbMATHGoogle 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 zbMATHGoogle Scholar
  5. 5.
    Turelli, M.: Theor. Popul. Biol. 12, 140 (1977) MathSciNetzbMATHCrossRefGoogle 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) zbMATHGoogle 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) MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. 25.
    Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967) zbMATHGoogle Scholar
  26. 26.
    Freidlin, M.: J. Stat. Phys. 117, 617 (2004) MathSciNetADSzbMATHCrossRefGoogle 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) zbMATHGoogle Scholar
  30. 30.
    Pavliotis, G., Stuart, A.: Multiscale Model. Simul. 4, 1 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Sigurgeirsson, H., Stuart, A.M.: Phys. Fluids 14, 4352 (2002) MathSciNetADSCrossRefGoogle Scholar

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© 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

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