Journal of Statistical Physics

, Volume 135, Issue 2, pp 261–277 | Cite as

A Gentle Stochastic Thermostat for Molecular Dynamics

  • Ben Leimkuhler
  • Emad Noorizadeh
  • Florian Theil


We discuss a dynamical technique for sampling the canonical measure in molecular dynamics. We present a method that generalizes a recently proposed scheme (Samoletov et al., J. Stat. Phys. 128:1321–1336, 2007), and which controls temperature by use of a device similar to that of Nosé dynamics, but adds random noise to improve ergodicity. In contrast to Langevin dynamics, where noise is added directly to each physical degree of freedom, the new scheme relies on an indirect coupling to a single Brownian particle. For a model with harmonic potentials, we show under a mild non-resonance assumption that we can recover the canonical distribution. In spite of its stochastic nature, experiments suggest that it introduces a relatively weak perturbative effect on the physical dynamics, as measured by perturbation of temporal autocorrelation functions. The kinetic energy is well controlled even in the early stages of a simulation.


Ergodicity Hypoellipticity Temperature control Molecular dynamics Nosé-Hoover thermostat 


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  1. 1.
    Barth, E., Leimkuhler, B., Sweet, C.: Approach to thermal equilibrium in biomolecular simulation. Lect. Notes Comput. Sci. Eng. 49, 125–140 (2005) CrossRefGoogle Scholar
  2. 2.
    Birkhoff, G.D.: Proof of the ergodic theorem. Proc. Natl. Acad. Sci. U.S.A. 17(12), 656 (1931) CrossRefADSGoogle Scholar
  3. 3.
    Bond, S.D., Leimkuhler, B.J.: Molecular dynamics and the accuracy of numerically computed averages. Acta Numer. 16, 1–65 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bou-Rabee, N., Owhadi, H.: Boltzmann-Gibbs preserving stochastic variational integrator (2007).
  5. 5.
    Brünger, A., Brooks, C.B., Karplus, M.: Stochastic boundary conditions for molecular dynamics simulations of st2 water. J. Chem. Phys. Lett. 105(5), 495–500 (1984) CrossRefADSGoogle Scholar
  6. 6.
    Bussi, G., Donadio, D., Parrinello, M.: Canonical sampling through velocity rescaling. J. Chem. Phys. 126, 014,101 (2007) CrossRefGoogle Scholar
  7. 7.
    Evans, D., Holian, B.: The Nosé-Hoover thermostat. J. Chem. Phys. 83, 4069–4074 (1985) CrossRefADSGoogle Scholar
  8. 8.
    Hairer, M., Mattingly, J.: Ergodicity of the 2d Navier-Stokes equations with degenerate stochastic forcing. Ann. Math. 164(3) (2006) Google Scholar
  9. 9.
    Helffer, B., Nier, F.: Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians. Springer, New York (2005) zbMATHGoogle Scholar
  10. 10.
    Hoover, W.: Canonical dynamics: equilibrium phase space distributions. Phys. Rev. A 31, 1695–1697 (1985) CrossRefADSGoogle Scholar
  11. 11.
    Hoover, W.G.: Molecular Dynamics. Springer, New York (1986) Google Scholar
  12. 12.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. Springer, New York (1985) Google Scholar
  13. 13.
    Ingrassia, S.: On the rate of convergence of the metropolis algorithm and Gibbs sampler by geometric bounds. Ann. Appl. Probab. 4(2), 347–389 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kaczmarski, M., Rurali, R., Hernández, E.: Reversible scaling simulations of the melting transition in silicon. Phys. Rev. B 69, 214,105 (2004) CrossRefGoogle Scholar
  15. 15.
    Khinchin, A.I.: Mathematical Foundations of Statistical Physics. Dover, New York (1949) Google Scholar
  16. 16.
    Legoll, F., Luskin, M., Moeckel, R.: Non-ergodicity of the Nosé-Hoover thermostatted harmonic oscillator. Arch. Ration. Mech. Anal. 184, 449–463 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Martyna, G.J., Klein, M.L., Tuckerman, M.: Nosé-Hoover chains: The canonical ensemble via continuous dynamics. J. Chem. Phys. 97(4), 2635–2643 (1992) CrossRefADSGoogle Scholar
  18. 18.
    Mattingly, J.C., Stuart, A.M.: Geometric ergodicity of some hypo-elliptic diffusions for particle motions. Markov Processes Relat. Fields 8(2), 199–214 (2002) zbMATHMathSciNetGoogle Scholar
  19. 19.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Melchionna, S.: Design of quasisymplectic propagators for Langevin dynamics. J. Chem. Phys. 127, 044,108 (2007) CrossRefGoogle Scholar
  21. 21.
    Meyn, S.P., Tweedie, R.: Markov Chains and Stochastic Stability. Springer, London (1993) zbMATHGoogle Scholar
  22. 22.
    Milstein, G., Tretyakov, N.: Quasi-symplectic methods for Langevin-type equations. IMA J. Numer. Anal. 23(3), 593–626 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Norris, J.: Simplified Malliavin calculus. In: Séminaire de Probabilités XX 1984/85. Lecture Notes in Mathematics, pp. 101–130. Springer, Berlin (1986) CrossRefGoogle Scholar
  24. 24.
    Nosé, S.: A unified formulation of the constant temperature molecular dynamics method. J. Chem. Phys. 81, 511–519 (1984) CrossRefADSGoogle Scholar
  25. 25.
    Penrose, O.: Foundations of Statistical Mechanics: a Deductive Treatment. Pergamon, Elmsford (1970) zbMATHGoogle Scholar
  26. 26.
    Petersen, K.: Ergodic Theory. Cambridge Studies in Advanced Mathematics, vol. 2. Cambridge University Press, Cambridge (1989) zbMATHGoogle Scholar
  27. 27.
    Quigley, D., Probert, M.: Langevin dynamics in constant pressure extended systems. J. Chem. Phys. 120, 11432 (2004) CrossRefADSGoogle Scholar
  28. 28.
    Roberts, G.O., Tweedie, R.L.: Exponential convergence of Langevin diffusions and their discrete approximations. Bernoulli 2(4), 341–363 (1995) CrossRefMathSciNetGoogle Scholar
  29. 29.
    Rosenthal, J.: Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Am. Stat. Assoc. 90(430), 558–566 (1995) zbMATHCrossRefGoogle Scholar
  30. 30.
    Samoletov, A., Chaplain, M.A.J., Dettmann, C.P.: Thermostats for “slow” configurational modes. J. Stat. Phys. 128, 1321–1336 (2007) zbMATHCrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Skeel, R.D., Izaguirre, J.A.: An impulse integrator for Langevin dynamics. Mol. Phys. 100, 3885 (2002) CrossRefADSGoogle Scholar
  32. 32.
    Vanden-Eijnden, E., Ciccotti, G.: Second-order integrators for Langevin equations with holonomic constraints. Chem. Phys. Lett. 429, 310–316 (2006) CrossRefADSGoogle Scholar
  33. 33.
    Villani, C.: Topics in optimal transportation. Am. Math. Soc. 58 (2003) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Ben Leimkuhler
    • 1
  • Emad Noorizadeh
    • 1
  • Florian Theil
    • 2
  1. 1.The Maxwell Institute and School of MathematicsUniversity of EdinburghEdinburghUK
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK

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