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A gentle stochastic thermostat for molecular dynamics. (English) Zbl 1179.82065

A dynamical technique for sampling the canonical measure in molecular dynamics is presented. The method generalizes the scheme by Samoletov, Chaplain and Dettmann, which aims to combine the advantages of the Langevin-thermostat with those of the Nosé-Hoover method. It can be viewed as a Nosé-Hoover method in which the thermostat variable is a Brownian particle. 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 examples of a harmonic oscillator and of three particle connected by springs, interacting with each other through a Lennard-Jones potential are investigated. The kinetic energy is well controlled even in the early stages of a simulation. The primary distinction between the method presented here and others in the literature is an analysis of ergodicity, making use of the concept of hypoellipticity with respect to the operator defining the right hand side of the Fokker-Planck equations. The new dynamics has an invariant probability measure, which is proportional to the Boltzmann-Gibbs distribution. It is proved analytically that under a non-resonance assumption, an open connected set U with full measure can be constructed, such that the probability measure is ergodic on U.

MSC:

82B30 Statistical thermodynamics
60J65 Brownian motion
81V55 Molecular physics
82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
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[1] Barth, E., Leimkuhler, B., Sweet, C.: Approach to thermal equilibrium in biomolecular simulation. Lect. Notes Comput. Sci. Eng. 49, 125–140 (2005) · Zbl 1094.92004 · doi:10.1007/3-540-31618-3_8
[2] Birkhoff, G.D.: Proof of the ergodic theorem. Proc. Natl. Acad. Sci. U.S.A. 17(12), 656 (1931) · Zbl 0003.25602 · doi:10.1073/pnas.17.12.656
[3] Bond, S.D., Leimkuhler, B.J.: Molecular dynamics and the accuracy of numerically computed averages. Acta Numer. 16, 1–65 (2007) · Zbl 1131.82019 · doi:10.1017/S0962492906280012
[4] Bou-Rabee, N., Owhadi, H.: Boltzmann-Gibbs preserving stochastic variational integrator (2007). http://arxiv.org/abs/0712.4123 · Zbl 1171.37027
[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) · doi:10.1016/0009-2614(84)80098-6
[6] Bussi, G., Donadio, D., Parrinello, M.: Canonical sampling through velocity rescaling. J. Chem. Phys. 126, 014,101 (2007) · doi:10.1063/1.2408420
[7] Evans, D., Holian, B.: The Nosé-Hoover thermostat. J. Chem. Phys. 83, 4069–4074 (1985) · doi:10.1063/1.449071
[8] Hairer, M., Mattingly, J.: Ergodicity of the 2d Navier-Stokes equations with degenerate stochastic forcing. Ann. Math. 164(3) (2006) · Zbl 1130.37038
[9] Helffer, B., Nier, F.: Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians. Springer, New York (2005) · Zbl 1072.35006
[10] Hoover, W.: Canonical dynamics: equilibrium phase space distributions. Phys. Rev. A 31, 1695–1697 (1985) · doi:10.1103/PhysRevA.31.1695
[11] Hoover, W.G.: Molecular Dynamics. Springer, New York (1986) · Zbl 0615.76004
[12] Hörmander, L.: The Analysis of Linear Partial Differential Operators. Springer, New York (1985) · Zbl 0601.35001
[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) · Zbl 0802.60061 · doi:10.1214/aoap/1177005064
[14] Kaczmarski, M., Rurali, R., Hernández, E.: Reversible scaling simulations of the melting transition in silicon. Phys. Rev. B 69, 214,105 (2004) · doi:10.1103/PhysRevB.69.214105
[15] Khinchin, A.I.: Mathematical Foundations of Statistical Physics. Dover, New York (1949) · Zbl 0037.41102
[16] Legoll, F., Luskin, M., Moeckel, R.: Non-ergodicity of the Nosé-Hoover thermostatted harmonic oscillator. Arch. Ration. Mech. Anal. 184, 449–463 (2007) · Zbl 1122.82002 · doi:10.1007/s00205-006-0029-1
[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) · doi:10.1063/1.463940
[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) · Zbl 1014.60059
[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) · Zbl 1075.60072 · doi:10.1016/S0304-4149(02)00150-3
[20] Melchionna, S.: Design of quasisymplectic propagators for Langevin dynamics. J. Chem. Phys. 127, 044,108 (2007) · doi:10.1063/1.2753496
[21] Meyn, S.P., Tweedie, R.: Markov Chains and Stochastic Stability. Springer, London (1993) · Zbl 0925.60001
[22] Milstein, G., Tretyakov, N.: Quasi-symplectic methods for Langevin-type equations. IMA J. Numer. Anal. 23(3), 593–626 (2003) · Zbl 1055.65141 · doi:10.1093/imanum/23.4.593
[23] Norris, J.: Simplified Malliavin calculus. In: Séminaire de Probabilités XX 1984/85. Lecture Notes in Mathematics, pp. 101–130. Springer, Berlin (1986)
[24] Nosé, S.: A unified formulation of the constant temperature molecular dynamics method. J. Chem. Phys. 81, 511–519 (1984) · doi:10.1063/1.447334
[25] Penrose, O.: Foundations of Statistical Mechanics: a Deductive Treatment. Pergamon, Elmsford (1970) · Zbl 0193.27801
[26] Petersen, K.: Ergodic Theory. Cambridge Studies in Advanced Mathematics, vol. 2. Cambridge University Press, Cambridge (1989) · Zbl 0676.28008
[27] Quigley, D., Probert, M.: Langevin dynamics in constant pressure extended systems. J. Chem. Phys. 120, 11432 (2004) · Zbl 1196.76057 · doi:10.1063/1.1755657
[28] Roberts, G.O., Tweedie, R.L.: Exponential convergence of Langevin diffusions and their discrete approximations. Bernoulli 2(4), 341–363 (1995) · Zbl 0870.60027 · doi:10.2307/3318418
[29] Rosenthal, J.: Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Am. Stat. Assoc. 90(430), 558–566 (1995) · Zbl 0824.60077 · doi:10.2307/2291067
[30] Samoletov, A., Chaplain, M.A.J., Dettmann, C.P.: Thermostats for ”slow” configurational modes. J. Stat. Phys. 128, 1321–1336 (2007) · Zbl 1128.82006 · doi:10.1007/s10955-007-9365-2
[31] Skeel, R.D., Izaguirre, J.A.: An impulse integrator for Langevin dynamics. Mol. Phys. 100, 3885 (2002) · doi:10.1080/0026897021000018321
[32] Vanden-Eijnden, E., Ciccotti, G.: Second-order integrators for Langevin equations with holonomic constraints. Chem. Phys. Lett. 429, 310–316 (2006) · doi:10.1016/j.cplett.2006.07.086
[33] Villani, C.: Topics in optimal transportation. Am. Math. Soc. 58 (2003) · Zbl 1106.90001
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