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Journal of Statistical Physics

, Volume 158, Issue 1, pp 1–36 | Cite as

Langevin Dynamics with Space-Time Periodic Nonequilibrium Forcing

  • R. Joubaud
  • G. A. Pavliotis
  • G. Stoltz
Article

Abstract

We present results on the ballistic and diffusive behavior of the Langevin dynamics in a periodic potential that is driven away from equilibrium by a space-time periodic driving force, extending some of the results obtained by Collet and Martinez in (J Math Biol, 56(6):765–792 2008). In the hyperbolic scaling, a nontrivial average velocity can be observed even if the external forcing vanishes in average. More surprisingly, an average velocity in the direction opposite to the forcing may develop at the linear response level—a phenomenon called negative mobility. The diffusive limit of the non-equilibrium Langevin dynamics is also studied using the general methodology of central limit theorems for additive functionals of Markov processes. To apply this methodology, which is based on the study of appropriate Poisson equations, we extend recent results on pointwise estimates of the resolvent of the generator associated with the Langevin dynamics. Our theoretical results are illustrated by numerical simulations of a two-dimensional system.

Keywords

Linear response of nonequilibrium systems Mobility Langevin dynamics Effective diffusion coefficient Hypoelliptic diffusions Langevin dynamics Linear response 

Notes

Acknowledgments

This work was initiated while GP was visiting the INRIA team MICMAC (now MATHERIALS) at CERMICS. The hospitality and financial support from INRIA are greatly acknowledged. RJ’s research is supported by the EPSRC through grant EP/J009636/1. GP’s research is partially supported by the EPSRC through grants EP/J009636/1 and EP/H034587/1. GS’s research is partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement number 614492. The authors benefited from discussions with Stefano Olla and Stephan De Bièvre.

References

  1. 1.
    Collet, P., Martínez, S.: Asymptotic velocity of one dimensional diffusions with periodic drift. J. Math. Biol. 56(6), 765–792 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Gammaitoni, L., Hanggi, P., Jung, P., Marchesoni, F.: Stochastic resonance. Rev. Mod. Phys. 70(1), 223–287 (1998)ADSCrossRefGoogle Scholar
  3. 3.
    Reimann, P.: Brownian motors: noisy transport far from equilibrium. Phys. Rep. 361(2–4), 57–265 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Machura, L., Kostur, M., Talkner, P., Łuczka, J., Hänggi, P.: Absolute negative mobility induced by thermal equilibrium fluctuations. Phys. Rev. Lett. 98, 040601 (2007)ADSCrossRefGoogle Scholar
  5. 5.
    Reimann, P., Van den Broeck, C., Linke, H., Rubi, J.M., Perez-Madrid, A.: Giant acceleration of free diffusion by use of tilted periodic potentials. Phys. Rev. Lett. 87(1), 010602 (2001)ADSCrossRefGoogle Scholar
  6. 6.
    Pavliotis, G.A.: A multiscale approach to Brownian motors. Phys. Lett. A 344, 331–345 (2005)ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. AMS Chelsea Publishing, New York (2011)zbMATHGoogle Scholar
  8. 8.
    Garnier, J.: Homogenization in a periodic and time-dependent potential. SIAM J. Appl. Math. 57(1), 95–111 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Kubo, R., Toda, M., Hashitsume, N.: Statistical Physics II. Nonequilibrium statistical mechanics 2nd (ed), Springer Series in Solid-State Sciences, vol. 31. Springer, Berlin (1991).Google Scholar
  10. 10.
    Resibois, P., De Leener, M.: Classical Kinetic Theory of Fluids. Wiley, New York (1977)Google Scholar
  11. 11.
    Joubaud, R., Stoltz, G.: Nonequilibrium shear viscosity computations with Langevin dynamics. Multiscale Model. Sim. 10, 191–216 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Komorowski, T., Olla, S.: On the superdiffusive behavior of passive tracer with a Gaussian drift. J. Stat. Phys. 108(3–4), 647–668 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Lebowitz, J.L., Rost, H.: The Einstein relation for the displacement of a test particle in a random environment. Stoch. Proc. Appl. 54(2), 183–196 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Rodenhausen, H.: Einstein’s relation between diffusion constant and mobility for a diffusion model. J. Stat. Phys. 55(5–6), 1065–1088 (1989)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Latorre, J.C., Pavliotis, G.A., Kramer, P.R.: Corrections to Einstein’s relation for Brownian motion in a tilted periodic potential. J. Stat. Phys. 150(4), 776–803 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Herrmann, S., Imkeller, P.: The exit problem for diffusions with time-periodic drift and stochastic resonance. Ann. Appl. Probab. 15(1), 39–68 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Komorowski, T., Olla, S.: On mobility and Einstein relation for tracers in time-mixing random environments. J. Stat. Phys. 118(3–4), 407–435 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Leimkuhler, B., Matthews, Ch., Stoltz, G.: The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics. arXiv:1308.5814 (2013)
  19. 19.
    Cattiaux, P., Chafaï, D., Guillin, A.: Central limit theorems for additive functionals of ergodic Markov diffusions processes. ALEA Lat Am. J Probab. Math. Stat. 9(2), 337–382 (2012)zbMATHMathSciNetGoogle Scholar
  20. 20.
    De Masi, A., Ferrari, P.A., Goldstein, S., Wick, W.D.: An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Stat. Phys. 55(3–4), 787–855 (1989)ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Kipnis, C., Varadhan, S.R.S.: Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys. 104, 1–19 (1986)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov processes: Time symmetry and martingale approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 345. Springer, Heidelberg (2012).Google Scholar
  23. 23.
    Talay, D.: Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Proc. Rel. Fields 8, 163–198 (2002)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Kopec, M.: Weak Backward Error Analysis for Langevin Process. arXiv:1310.2599 (2013)
  25. 25.
    Hairer, M., Pavliotis, G.A.: Periodic homogenization for hypoelliptic diffusions. J. Stat. Phys. 117(1/2), 261–279 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Pavliotis, G.A.: Asymptotic analysis of the Green-Kubo formula. IMA J. Appl. Math. 75(6), 951–967 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Pavliotis, G.A.: Homogenization Theory for Advection-Diffusion Equations with Mean Flow, Ph.D Thesis. Rensselaer Polytechnic Institute, Troy, NY (2002).Google Scholar
  28. 28.
    Hairer, M., Pavliotis, G.A.: From ballistic to diffusive behavior in periodic potentials. J. Stat. Phys. 131(1), 175–202 (2008)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Papanicolaou, G.C., Varadhan, S.R.S.: Ornstein-Uhlenbeck process in a random potential. Commun. Pure Appl. Math. 38(6), 819–834 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Pavliotis, G.A., Vogiannou, A.: Diffusive transport in periodic potentials: underdamped dynamics. Fluct. Noise Lett. 8(2), L155–173 (2008)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Höpfner, R., Kutoyants, Y.: Estimating discontinuous periodic signals in a time inhomogeneous diffusion. Stat. Inference Stoch. Process. 13(3), 193–230 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Hairer, M., Mattingly, J.C.: Yet another look at Harris’ ergodic theorem for Markov chains. Seminar on Stochastic Analysis, Random Fields and Applications VI. Progress in Probability, vol. 63, pp. 109–117. Birkhäuser/Springer, Basel (2011).Google Scholar
  33. 33.
    Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability 2nd (ed.). Cambridge University Press, Cambridge (2009).Google Scholar
  34. 34.
    Tierney, L.: Markov chains for exploring posterior distributions. Ann. Statist. 22(4), 1701–1762 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Nummelin, E.: General Irreducible Markov Chains and Nonnegative Operators. Cambridge University Press, Cambridge (1984)CrossRefGoogle Scholar
  36. 36.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, 2nd (ed), vol. 113. Springer, Berlin (1991).Google Scholar
  37. 37.
    Rey-Bellet, L.: Ergodic properties of markov processes. In: Attal, S., Joye, A., Pillet, C.A. (eds.) Open Quantum Systems II. Lecture Notes in Mathematics, vol. 1881, pp. 1–39. Springer, Berlin (2006)CrossRefGoogle Scholar
  38. 38.
    Mattingly, J.C., Stuart, A.M., Higham, D.J.: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stoch. Proc. Appl. 101(2), 185–232 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Stoltz, G.: Molecular Simulation: Nonequilibrium and Dynamical Problems. Habilitation thesis. Université Paris Est. http://tel.archives-ouvertes.fr/tel-00709965 (2012)
  40. 40.
    Eckmann, J.-P., Hairer, M.: Spectral properties of hypoelliptic operators. Commun. Math. Phys. 235(2), 233–253 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Hérau, F., Nier, F.: Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch. Ration. Mech. Anal. 171, 151–218 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Villani, C.: Hypocoercivity, vol. 202, no. 950. Memoirs of the American Mathematical Society (2009)Google Scholar
  43. 43.
    Ethier, S.N., Kurtz, T.G.: Markov Pocesses. Probability and mathematical statistics. Wiley series in probability and mathematical statistics. Wiley, New York (1986).Google Scholar
  44. 44.
    Billingsley, P.: Convergence of Probability Measures. Wiley series in probability and statistics. Wiley, Hoboken, NJ (1999)CrossRefzbMATHGoogle Scholar
  45. 45.
    Billingsley, P.: Probability and Measure. Wiley series in probability and statistics. Wiley, Hoboken, NJ (1995)zbMATHGoogle Scholar
  46. 46.
    Helland, I.S.: Central limit theorems for martingales with discrete or continuous time. Scand. J. Stat. 9, 79–94 (1982)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsImperical College LondonLondonUK
  2. 2.CERMICS (ENPC)INRIA, Université Paris-EstMarne-la-ValléeFrance

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