Skip to main content
Log in

The Smoluchowski-Kramers Limit of Stochastic Differential Equations with Arbitrary State-Dependent Friction

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study a class of systems of stochastic differential equations describing diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe their dynamics in the small mass limit. Our systems have arbitrary state-dependent friction and noise coefficients. We identify the limiting equation and, in particular, the additional drift term that appears in the limit is expressed in terms of the solution to a Lyapunov matrix equation. The proof uses a theory of convergence of stochastic integrals developed by Kurtz and Protter. The result is sufficiently general to include systems driven by both white and Ornstein–Uhlenbeck colored noises. We discuss applications of the main theorem to several physical phenomena, including the experimental study of Brownian motion in a diffusion gradient.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bellman, R.: Introduction to Matrix Analysis, Volume 19 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. Reprint of the second edition, With a foreword by Gene Golub (1970)

  2. Blount D.: Comparison of stochastic and deterministic models of a linear chemical reaction with diffusion. Ann. Probab. 19(4), 1440–1462 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brettschneider T., Volpe G., Helden L., Wehr J., Bechinger C.: Force measurement in the presence of brownian noise: equilibrium-distribution method versus drift method. Phys. Rev. E 83, 041113 (2011)

    Article  ADS  Google Scholar 

  4. Cerrai S., Freidlin M.: Small mass asymptotics for a charged particle in a magnetic field and long-time influence of small perturbations. J. Stat. Phys. 144, 101–123 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Duhr S., Braun D.: Why molecules move along a temperature gradient. Proc. Natl. Acad. Sci. USA 103, 19678–19682 (2006)

    Article  ADS  Google Scholar 

  6. Freidlin M.: Some remarks on the Smoluchowski-Kramers approximation. J. Stat. Phys. 117, 617–634 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Freidlin M., Hu W.: Smoluchowskikramers approximation in the case of variable friction. J. Math. Sci. 179, 184–207 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Freidlin M., Hu W., Wentzell A.: Small mass asymptotic for the motion with vanishing friction. Stoch. Process. Appl. 123, 45–75 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Freidlin M., Weber M.: Perturbations of the motion of a charged particle in a noisy magnetic field. J. Stat. Phys. 147, 565–581 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Hänggi P.: Nonlinear fluctuations: the problem of deterministic limit and reconstruction of stochastic dynamics. Phys. Rev. A 25, 1130–1136 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  11. Happel J., Brenner H.: Low Reynolds number hydrodynamics with special applications to particulate media. Prentice-Hall Inc., Englewood Cliffs (1965)

    Google Scholar 

  12. Hottovy S., Volpe G., Wehr J.: Noise-induced drift in stochastic differential equations with arbitrary friction and diffusion in the Smoluchowski-Kramers limit. J. Stat. Phys. 146, 762–773 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hottovy S., Volpe G., Wehr J.: Thermophoresis of Brownian particles driven by coloured noise. EPL (Europhys. Lett.) 99, 60002 (2012)

    Article  ADS  Google Scholar 

  14. Karatzas I., Shreve S.E.: Brownian Motion and Stochastic Calculus, Volume 113 of Graduate Texts in Mathematics, second edition. Springer, New York (1991)

    Google Scholar 

  15. Kramers H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Kupferman R., Pavliotis G.A., Stuart A.M.: Itô versus Stratonovich white-noise limits for systems with inertia and colored multiplicative noise. Phys. Rev. E 70, 036120 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  17. Kurtz T.G., Protter P.: Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19, 1035–1070 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kwon C., Ao P., Thouless D.J.: Structure of stochastic dynamics near fixed points. Proc. Natl. Acad. Sci. USA 102, 13029–13033 (2005)

    Article  ADS  Google Scholar 

  19. Nelson E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)

    MATH  Google Scholar 

  20. Øksendal B.: Stochastic Differential Equations: An Introduction with Applications, Universitext, Sixth edition. Springer, Berlin (2003)

    Book  Google Scholar 

  21. Ortega J.M.: Matrix theory. The University Series in Mathematics. Plenum Press, New York (1987) A second course

    Google Scholar 

  22. Papanicolaou A.: Filtering for fast mean-reverting processes. Asymptot. Anal. 70, 155–176 (2010)

    MATH  MathSciNet  Google Scholar 

  23. Papanicolaou, G.C.: Introduction to the asymptotic analysis of stochastic equations. In: Modern Modeling of Continuum Phenomena (Ninth Summer Sem. Appl. Math., Rensselaer Polytech. Inst., Troy, N.Y., 1975). Lectures in Appl. Math., vol. 16. Amer. Math. Soc., pp. 109–147, Providence, R.I., (1977)

  24. Pardoux È., Veretennikov AY: On Poisson equation and diffusion approximation. I,II,III. Ann. Probab. 31, 1166–1192 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Pavliotis, G.A.: Multiscale Methods Volume 53 of Texts in Applied Mathematics. Springer, New York. Averaging and homogenization (2008)

  26. Pesce G., Volpe G., De Luca A.C., Rusciano G., Volpe G.: Quantitative assessment of non-conservative radiation forces in an optical trap. EPL (Europhys. Lett.) 86, 38002 (2009)

    Article  ADS  Google Scholar 

  27. Piazza R.: Thermophoresis: moving particles with thermal gradients. Soft Matt. 4, 1740–1744 (2008)

    Article  ADS  Google Scholar 

  28. Protter P.E.: Stochastic Integration and Differential Equations, Volume 21 of Stochastic Modelling and Applied Probability, Second edition. Springer, Berlin (2005) Version 2.1, Corrected third printing

    Google Scholar 

  29. Revuz D., Yor M.: Continuous Martingales and Brownian Motion, Volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], third edition. Springer, Berlin (1999)

    Google Scholar 

  30. Sancho J.M., San Miguel M., Dürr M.: Adiabatic elimination for systems of Brownian particles with nonconstant damping coefficients. J. Stat. Phys. 28, 291–305 (1982)

    Article  ADS  MATH  Google Scholar 

  31. Schuss Z.: Theory and Applications of Stochastic Differential Equations, Wiley Series in Probability and Statistics. Wiley, New York (1980)

    Google Scholar 

  32. Shi J., Chen T., Yuan R., Yuan B., Ao P.: Relation of a new interpretation of stochastic differential equations to ito process. J. Stat. Phys. 148, 579–590 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. Simpson N.B., Dholakia K., Allen L., Padgett M.J.: Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner. Opt. Lett. 22, 52–54 (1997)

    Article  ADS  Google Scholar 

  34. Smoluchowski M.: Drei vortrage über diffusion brownsche bewegung and koagulation von kolloidteilchen. Phys. Z. 17, 557–585 (1916)

    ADS  Google Scholar 

  35. Toda M., Kubo R., Saitô N.: Statistical physics. I. Equilibrium statistical mechanics, volume 30 of Springer Series in Solid-State Sciences, second edition. Springer, Berlin (1992)

    Google Scholar 

  36. Volpe G., Helden L., Brettschneider T., Wehr J., Bechinger C.: Influence of noise on force measurements. Phys. Rev. Lett. 104, 170602 (2010)

    Article  ADS  Google Scholar 

  37. Volpe G., Volpe G., Petrov D.: Singular-point characterization in microscopic flows. Phys. Rev. E 77, 037301 (2008)

    Article  ADS  Google Scholar 

  38. Williams D.: Probability with martingales. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  39. Wong E., Zakai M.: On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat. 36, 1560–1564 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  40. Zwanzig R.: Nonequilibrium Statistical Mechanics. Oxford University Press, Oxford (2001)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Scott Hottovy.

Additional information

Communicated by H. Spohn

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hottovy, S., McDaniel, A., Volpe, G. et al. The Smoluchowski-Kramers Limit of Stochastic Differential Equations with Arbitrary State-Dependent Friction. Commun. Math. Phys. 336, 1259–1283 (2015). https://doi.org/10.1007/s00220-014-2233-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-2233-4

Keywords

Navigation