Journal of Statistical Physics

, Volume 143, Issue 4, pp 715–724 | Cite as

The Langevin Limit of the Nosé-Hoover-Langevin Thermostat

  • Jason Frank
  • Georg A. Gottwald


In this note we study the asymptotic limit of large variance in a stochastically perturbed thermostat model, the Nosé-Hoover-Langevin device. We show that in this limit, the model reduces to a Langevin equation with one-dimensional Wiener process, and that the perturbation is in the direction of the conjugate momentum vector. Numerical experiments with a double well potential corroborate the asymptotic analysis.


Thermostat methods Nosé-Hoover dynamics Langevin dynamics Homogenization methods Canonical sampling 


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  1. 1.
    Bulgac, A., Kusnezov, D.: Canonical ensemble averages from pseudomicrocanonical dynamics. Phys. Rev. A 42, 5045–5048 (1990) ADSCrossRefGoogle Scholar
  2. 2.
    Bussi, G., Parinello, M.: Stochastic thermostats: comparison of local and global schemes. Comput. Phys. Commun. 179, 26–29 (2008) ADSCrossRefGoogle Scholar
  3. 3.
    Bussi, G., Donadio, D., Parinello, M.: Canonical sampling through velocity rescaling. J. Chem. Phys. 126, 014101 (2007) ADSCrossRefGoogle Scholar
  4. 4.
    Dubinkina, S., Frank, J., Leimkuhler, B.: Simplified modelling of a thermal bath, with application to a fluid vortex system. SIAM Multiscale Model. Simul. 8, 1882–1901 (2010) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Higham, D.J., Mao, X., Stuart, A.M.: Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40, 1041–1063 (2002) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Hoover, W.: Canonical dynamics: equilibrium phase space distributions. Phys. Rev. A 31, 1695–1697 (1985) ADSCrossRefGoogle Scholar
  7. 7.
    Khasminsky, R.Z.: On stochastic processes defined by differential equations with a small parameter. Theory Probab. Appl. 11, 211–228 (1966) CrossRefGoogle Scholar
  8. 8.
    Kurtz, T.G.: A limit theorem for perturbed operator semigroups with applications to random evolutions. J. Funct. Anal. 12, 55–67 (1973) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Kurtz, T.G.: Limit theorems and diffusion approximations for density dependent Markov chains. Math. Program. Stud. 5, 67–78 (1976) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kurtz, T.G.: Strong approximation theorems for density dependent Markov chains. Stoch. Process. Appl. 6, 223–240 (1978) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Leimkuhler, B., Noorizadeh, E., Theil, F.: A gentle stochastic thermostat for molecular dynamics. J. Stat. Phys. 135, 261–277 (2009) MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    Leimkuhler, B., Noorizadeh, E., Penrose, O.: Comparing the efficiencies of stochastic isothermal molecular dynamics methods. J. Stat. Phys. (2011, to appear) Google Scholar
  13. 13.
    Legoll, F., Luskin, M., Moeckel, R.: Non-ergodicity of Nosé-Hoover thermostatted harmonic oscillator. Arch. Ration. Mech. Anal. 184, 449–463 (2007) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Legoll, F., Luskin, M., Moeckel, R.: Non-ergodicity of Nosé-Hoover dynamics. Nonlinearity 22, 1673–1694 (2009) MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    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, 185–232 (2002) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Nosé, S.: A molecular dynamics methods for simulations in the canonical ensemble. Mol. Phys. 52, 255–268 (1984) ADSCrossRefGoogle Scholar
  17. 17.
    Nosé, S.: A unified formulation of the constant temperature molecular dynamics method. J. Chem. Phys. 81, 511–519 (1984) ADSCrossRefGoogle Scholar
  18. 18.
    Papanicolaou, G.C.: Introduction to the asymptotic analysis of stochastic equations. In: DiPrima, R.C. (ed.) Modern Modeling of Continuum Phenomena. AMS, Providence (1974) Google Scholar
  19. 19.
    Pavliotis, G.A., Stuart, A.M.: Multiscale Methods—Averaging and Homogenization. Texts in Applied Mathematics, vol. 53. Springer, New York (2008) MATHGoogle Scholar
  20. 20.
    Samoletov, A., Chaplain, M.A.J., Dettmann, C.P.: Thermostats for “slow” configurational modes. J. Stat. Phys. 128, 1321–1336 (2007) MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    Givon, D., Kupferman, R., Stuart, A.: Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity 17, R55–R127 (2004) MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands
  2. 2.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

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