Ergodic Properties of Quasi-Markovian Generalized Langevin Equations with Configuration Dependent Noise and Non-conservative Force

  • Benedict LeimkuhlerEmail author
  • Matthias SachsEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 282)


We discuss the ergodic properties of quasi-Markovian stochastic differential equations, providing general conditions that ensure existence and uniqueness of a smooth invariant distribution and exponential convergence of the evolution operator in suitably weighted \(L^{\infty }\) spaces, which implies the validity of central limit theorem for the respective solution processes. The main new result is an ergodicity condition for the generalized Langevin equation with configuration-dependent noise and (non-)conservative force.


Generalized Langevin equation Heat-bath Quasi-Markovian model Sampling Molecular dynamics Ergodicity Central limit theorem Non-equilibrium Mori-Zwanzig formalism Reduced model 



The authors wish to thank Greg Pavliotis (Imperial), Jonathan Mattingly (Duke) and Gabriel Stoltz (ENPC) for their generous assistance in providing comments at various stages of this project. In particular, the authors thank Jonathan Mattingly for pointing out the possibility of using Girsanov’s theorem in the proof of Lemma 7. Both authors acknowledge the support of the European Research Council (Rule Project, grant no. 320823). BJL further acknowledges the support of the EPSRC (grant no. EP/P006175/1) during the preparation of this article. The work of MS was supported by the National Science Foundation under grant DMS-1638521 to the Statistical and Applied Mathematical Sciences Institute.


  1. 1.
    Adelman, S., Doll, J.: Generalized Langevin equation approach for atom/solid-surface scattering: general formulation for classical scattering off harmonic solids. J. Chem. Phys. 64(6), 2375–2388 (1976)CrossRefGoogle Scholar
  2. 2.
    Bhattacharya, R.N.: On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 60(2), 185–201 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Carmona, P.: Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths. Stochast. Process. Appl. 117(8), 1076–1092 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ceriotti, M.: GLE4MD.
  5. 5.
    Ceriotti, M., Bussi, G., Parrinello, M.: Langevin equation with colored noise for constant-temperature molecular dynamics simulations. Phys. Rev. Lett. 102(2), 020601 (2009)CrossRefGoogle Scholar
  6. 6.
    Ceriotti, M., Bussi, G., Parrinello, M.: Colored-noise thermostats à la carte. J. Chem. Theory Comput. 6(4), 1170–1180 (2010)CrossRefGoogle Scholar
  7. 7.
    Darve, E., Solomon, J., Kia, A.: Computing generalized Langevin equations and generalized Fokker-Planck equations. Proc. Nat. Acad. Sci. 106(27), 10884–10889 (2009)CrossRefGoogle Scholar
  8. 8.
    Doll, J.D., Dion, D.R.: Generalized Langevin equation approach for atom/solid-surface scattering: numerical techniques for Gaussian generalized Langevin dynamics. J. Chem. Phys. 65(9), 3762–3766 (1976)CrossRefGoogle Scholar
  9. 9.
    Dym, H., McKean, H.P.: Gaussian Processes, Function Theory, and the Inverse Spectral Problem. Courier Corporation (2008)Google Scholar
  10. 10.
    Eckmann, J.-P., Hairer, M.: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212(1), 105–164 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Entropy production in nonlinear, thermally driven Hamiltonian systems. J. Stat. Phys. 95(1), 305–331 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Commun. Math. Phys. 201(3), 657–697 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Ford, G., Kac, M., Mazur, P.: Statistical mechanics of assemblies of coupled oscillators. J. Math. Phys. 6(4), 504–515 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Givon, D., Kupferman, R., Hald, O.H.: Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism. Isr. J. Math. 145(1), 221–241 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Givon, D., Kupferman, R., Stuart, A.: Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity 17(6), R55–R127 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Hairer, M., Mattingly, J.C.: Yet another look at Harris Ergodic theorem for Markov chains. In: Seminar on Stochastic Analysis, Random Fields and Applications VI, vol. 63, pp. 109–117. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    Hänggi, P.: Generalized Langevin equations: a useful tool for the perplexed modeller of nonequilibrium fluctuations? In: Lutz, S.-G., Thorsten, P. (eds.) Stochastic Dynamics, pp. 15–22. Springer, Heidelberg (1997)Google Scholar
  18. 18.
    Harris, T.E.: The existence of stationary measures for certain Markov processes. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 113–124 (1956)Google Scholar
  19. 19.
    Hohenegger, C., McKinley, S.A.: Fluid-particle dynamics for passive tracers advected by a thermally fluctuating viscoelastic medium. J. Comput. Phys. 340, 688–711 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Hörmander, L.: The analysis of linear partial differential operators III. Grundlehren der Mathematischen Wissenschaften [fundamental principles of mathematical sciences], vol. 274 (1985)Google Scholar
  21. 21.
    Jakišć, V., Pillet, C.-A.: Ergodic properties of the non-Markovian Langevin equation. Lett. Math. Phys. 41(1), 49–57 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Jakšić, V., Pillet, C.-A.: Spectral theory of thermal relaxation. J. Math. Phys. 38(4), 1757–1780 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Jakšić, V., Pillet, C.-A.: Ergodic properties of classical dissipative systems I. Acta Math. 181(2), 245–282 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Joubaud, R., Pavliotis, G., Stoltz, G.: Langevin dynamics with space-time periodic nonequilibrium forcing. J. Stat. Phys. 158(1), 1–36 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Kantorovich, L.: Generalized Langevin equation for solids. I. Rigorous derivation and main properties. Phys. Rev. B 78(9), 094304 (2008)Google Scholar
  26. 26.
    Kliemann, W.: Recurrence and invariant measures for degenerate diffusions. Ann. Probab. 15, 690–707 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Kupferman, R.: Fractional kinetics in Kac-Zwanzig heat bath models. J. Stat. Phys. 114(1), 291–326 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Kupferman, R., Stuart, A., Terry, J., Tupper, P.: Long-term behaviour of large mechanical systems with random initial data. Stoch. Dyn. 2(4), 533–562 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Lampo, T.J., Kuwada, N.J., Wiggins, P.A., Spakowitz, A.J.: Physical modeling of chromosome segregation in Escherichia coli reveals impact of force and DNA relaxation. Biophys. J. 108(1), 146–153 (2015)CrossRefGoogle Scholar
  30. 30.
    Lei, H., Baker, N.A., Li, X.: Data-driven parameterization of the generalized Langevin equation. Proc. Nat. Acad. Sci. 113(50), 14183–14188 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Leimkuhler, B., Matthews, C., Stoltz, G.: The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics. IMA J. Numer. Anal. 36(1), 13–79 (2016)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Lelièvre, T., Stoltz, G.: Partial differential equations and stochastic methods in molecular dynamics. Acta Numer. 25, 681–880 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Li, Z., Bian, X., Li, X., Karniadakis, G.E.: Incorporation of memory effects in coarse-grained modeling via the Mori-Zwanzig formalism. J. Chem. Phys. 143(24), 243128 (2015)CrossRefGoogle Scholar
  34. 34.
    Li, Z., Lee, H.S., Darve, E., Karniadakis, G.E.: Computing the non-Markovian coarse-grained interactions derived from the Mori-Zwanzig formalism in molecular systems: application to polymer melts. J. Chem. Phys. 146(1), 014104 (2017)CrossRefGoogle Scholar
  35. 35.
    Lim, S.H., Wehr, J.: Homogenization for a class of generalized Langevin equations with an application to thermophoresis. J. Stat. Phys. 174, 656–691 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Mattingly, J.C., Stuart, A.M., Higham, D.J.: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stoch. Process. Appl. 101(2), 185–232 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes I: criteria for discrete-time chains. Adv. Appl. Probab. 24(3), 542–574 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes II: continuous-time processes and sampled chains. Adv. Appl. Probab. 25(3), 487–517 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Springer Science & Business Media, Heidelberg (2012)Google Scholar
  40. 40.
    Mori, H.: A continued-fraction representation of the time-correlation functions. Prog. Theor. Phys. 34(3), 399–416 (1965)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Morriss, G.P., Evans, D.J.: Statistical Mechanics of Nonequilbrium Liquids. ANU Press, Canberra (2013)Google Scholar
  42. 42.
    Morrone, J.A., Markland, T.E., Ceriotti, M., Berne, B.: Efficient multiple time scale molecular dynamics: using colored noise thermostats to stabilize resonances. J. Chem. Phys. 134(1), 014103 (2011)CrossRefGoogle Scholar
  43. 43.
    Ness, H., Stella, L., Lorenz, C., Kantorovich, L.: Applications of the generalized Langevin equation: towards a realistic description of the baths. Phys. Rev. B 91(1), 014301 (2015)CrossRefGoogle Scholar
  44. 44.
    Ness, H., Genina, A., Stella, L., Lorenz, C.D., Kantorovich, L.: Nonequilibrium processes from generalized Langevin equations: realistic nanoscale systems connected to two thermal baths. Phys. Rev. B 93(17), 174303 (2016)CrossRefGoogle Scholar
  45. 45.
    Øksendal, B.: Stochastic differential equations. In: Stochastic Differential Equations, pp. 65–84. Springer, Heidelberg (2003)Google Scholar
  46. 46.
    Ottobre, M., Pavliotis, G.: Asymptotic analysis for the generalized Langevin equation. Nonlinearity 24(5), 1629–1653 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Pavliotis, G.A.: Stochastic Processes and Applications. Springer, Heidelberg (2016)Google Scholar
  48. 48.
    Redon, S., Stoltz, G., Trstanova, Z.: Error analysis of modified Langevin dynamics. J. Stat. Phys. 164(4), 735–771 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Rey-Bellet, L.: Statistical mechanics of anharmonic lattices. In: Advances in Differential Equations and Mathematical Physics (Birmingham, AL, 2002), vol. 327, pp. 283–293 (2003)Google Scholar
  50. 50.
    Rey-Bellet, L.: Ergodic properties of Markov processes. In: Open Quantum Systems II. Lecture Notes in Mathematics, vol. 1881, pp. 1–39. Springer, Heidelberg (2006)Google Scholar
  51. 51.
    Rey-Bellet, L.: Open classical systems. In: Open Quantum Systems II. Lecture Notes in Mathematics, vol. 1881, pp. 41–78. Springer, Heidelberg (2006)Google Scholar
  52. 52.
    Rey-Bellet, L., Thomas, L.E.: Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Commun. Math. Phys. 225(2), 305–329 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Rudin, W.: Fourier Analysis on Groups. Courier Dover Publications, New York (2017)zbMATHGoogle Scholar
  54. 54.
    Sachs, M., Leimkuhler, B., Danos, V.: Langevin dynamics with variable coefficients and nonconservative forces: from stationary states to numerical methods. Entropy 19(12), 647 (2017)CrossRefGoogle Scholar
  55. 55.
    Stein, M.L.: Interpolation of Spatial Data: Some Theory for Kriging. Springer Science & Business Media, Heidelberg (2012)Google Scholar
  56. 56.
    Stella, L., Lorenz, C., Kantorovich, L.: Generalized Langevin equation: an efficient approach to nonequilibrium molecular dynamics of open systems. Phys. Rev. B 89(13), 134303 (2014)CrossRefGoogle Scholar
  57. 57.
    Stroock, D.W., Varadhan, S.R.: On the support of diffusion processes with applications to the strong maximum principle. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (University of California, Berkeley, California, 1970/1971), vol. 3, pp. 333–359 (1972)Google Scholar
  58. 58.
    Talay, D.: Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Process. Related Fields 8(2), 163–198 (2002)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Villani, C.: Hypocoercivity. American Mathematical Society, Providence (2009)zbMATHCrossRefGoogle Scholar
  60. 60.
    Wu, X., Brooks, B.R., Vanden-Eijnden, E.: Self-guided Langevin dynamics via generalized Langevin equation. J. Comput. Chem. 37(6), 595–601 (2016)CrossRefGoogle Scholar
  61. 61.
    Zhang, F. (ed.): The Schur Complement and Its Applications. Numerical Methods and Algorithms, vol. 4. Springer Science & Business Media, Heidelberg (2006)Google Scholar
  62. 62.
    Zwanzig, R.: Memory effects in irreversible thermodynamics. Phys. Rev. 124(4), 983–992 (1961)zbMATHCrossRefGoogle Scholar
  63. 63.
    Zwanzig, R.: Nonlinear generalized Langevin equations. J. Stat. Phys. 9(3), 215–220 (1973)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.The School of Mathematics and the Maxwell Institute of Mathematical Sciences, James Clerk Maxwell BuildingUniversity of EdinburghEdinburghUK
  2. 2.Department of MathematicsDuke UniversityDurhamUSA
  3. 3.The Statistical and Applied Mathematical Sciences Institute (SAMSI)DurhamUSA

Personalised recommendations