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.
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References
Øksendal, B.: Stochastic Differential Equations. Springer, Berlin (1998)
Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. Springer, New York (1998)
Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1981)
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
Turelli, M.: Theor. Popul. Biol. 12, 140 (1977)
Ao, P.: Commun. Theor. Phys. 49, 1073 (2008)
Gardiner, C.: Handbook of Stochastic Methods. Springer, Berlin (1985)
Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1999)
Sussmann, H.: Ann. Probab., 19–41 (1978)
Ermark, D.L., McCammon, J.A.: J. Chem. Phys. 69, 1352 (1978)
Lançon, P., Batrouni, G., Lobry, L., Ostrowsky, N.: EPL (Europhys. Lett.) 54, 28 (2001)
Lau, A.W.C., Lubensky, T.C.: Phys. Rev. E 76, 011123 (2007)
Volpe, G., Helden, L., Brettschneider, T., Wehr, J., Bechinger, C.: Phys. Rev. Lett. 104, 170602 (2010)
Brettschneider, T., Volpe, G., Helden, L., Wehr, J., Bechinger, C.: Phys. Rev. E 83, 041113 (2011)
Wehr, J.: University of Arizona (2011, in preparation)
Ao, P., Kwon, C., Qian, H.: Complexity 12, 19 (2007)
Papanicolaou: In: AMS Seminar, Lecture Notes. Rensselaer Polytechnic Institute, Troy (1975).
Pavliotis, G.A., Stuart, A.M.: Multiscale Methods: Averaging and Homogenization. Springer, Berlin (2008)
Schuss, Z.: Theory and Application of Stochastic Differential Equations. Wiley, New York (1980)
Pardoux, È., Veretennikov, A.: Ann. Probab. 29 (2001)
Pardoux, È., Veretennikov, A.: Ann. Probab. 31 (2003)
Pardoux, È., Veretennikov, A.: Ann. Probab. 33 (2005)
Smoluchowski, M.: Phys. Z. 17, 557 (1916)
Kramers, H.: Physica 7, 284 (1940)
Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)
Freidlin, M.: J. Stat. Phys. 117, 617 (2004)
Kupferman, R., Pavliotis, G.A., Stuart, A.M.: Phys. Rev. E 70, 036120 (2004)
Courant, R., Hilbert, D.: Methods of Mathematical Physics. Interscience, New York (1953)
Lax, P.: Functional Analysis. Wiley, New York (2002)
Pavliotis, G., Stuart, A.: Multiscale Model. Simul. 4, 1 (2005)
Sigurgeirsson, H., Stuart, A.M.: Phys. Fluids 14, 4352 (2002)
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S.H. was supported by the VIGRE grant through the University of Arizona Applied Mathematics Program. J.W. was partially supported by the NSF grant DMS 1009508.
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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). https://doi.org/10.1007/s10955-012-0418-9
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DOI: https://doi.org/10.1007/s10955-012-0418-9