Journal of Statistical Physics

, Volume 97, Issue 3–4, pp 687–723

Analysis and Experiments for a Computational Model of a Heat Bath

  • A. M. Stuart
  • J. O. Warren
Article

Abstract

A question of some interest in computational statistical mechanics is whether macroscopic quantities can be accurately computed without detailed resolution of the fastest scales in the problem. To address this question a simple model for a distinguished particle immersed in a heat bath is studied (due to Ford and Kac). The model yields a Hamiltonian system of dimension 2N+2 for the distinguished particle and the degrees of freedom describing the bath. It is proven that, in the limit of an infinite number of particles in the heat bath (N→∞), the motion of the distinguished particle is governed by a stochastic differential equation (SDE) of dimension 2. Numerical experiments are then conducted on the Hamiltonian system of dimension 2N+2 (N≫1) to investigate whether the motion of the distinguished particle is accurately computed (i.e., whether it is close to the solution of the SDE) when the time step is small relative to the natural time scale of the distinguished particle, but the product of the fastest frequency in the heat bath and the time step is not small—the underresolved regime in which many computations are performed. It is shown that certain methods accurately compute the limiting behavior of the distinguished particle, while others do not. Those that do not are shown to compute a different, incorrect, macroscopic limit.

computational statistical mechanics molecular dynamics Hamiltonian systems stiff oscillatory systems stochastic differential equations Langevin equation symplectic methods energy-conserving methods 

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REFERENCES

  1. 1.
    U. Ascher and S. Reich, On some difficulties in integrating highly oscillatory Hamiltonian systems, Proc. Alg. for Macromolecular Modelling, 1997.Google Scholar
  2. 2.
    U. Ascher and S. Reich, The midpoint scheme and variants for Hamiltonian systems: Advantages and pitfalls, SIAM J. Sci. Comp., to appear.Google Scholar
  3. 3.
    F. Bornemann and Schütte, Homogenization of Hamiltonian systems with a strong constraining potential, Physica D 102:57–77 (1997).Google Scholar
  4. 4.
    A. J. Chorin, A. Kast, and R. Kupferman, On the prediction of large-scale dynamics using underresolved computations. Submitted to AMS Contemporary Mathematics, 1998.Google Scholar
  5. 5.
    A. J. Chorin, A. Kast, and R. Kupferman, Unresolved computation and optimal predictions, Comm. Pure Appl. Math., to appear.Google Scholar
  6. 6.
    A. J. Chorin, A. Kast, and R. Kupferman, Optimal prediction of underresolved dynamics, Proc. Nat. Acad. Sci. USA 95:4094–4098 (1998).Google Scholar
  7. 7.
    B. Cano, A. Stuart, E. Süli, and J. Warren, Stiff oscillatory systems, delta jumps and white noise, Technical Report SCCM-99-01, http://www-sccm.stanford.edu/pub/sccm/sccm-99-01.ps.gz.Google Scholar
  8. 8.
    G. W. Ford, J. T. Lewis, and R. F. O'Connell, Quantum Langevin equation, Phys. Rev. A 37:4419–4428 (1988).Google Scholar
  9. 9.
    G. W. Ford and M. Kac, On the quantum Langevin equation, J. Stat. Phys. 46:803–810 (1987).Google Scholar
  10. 10.
    O. Gonzalez and J. C. Simo, On the stability of symplectic and energy-momentum algorithms for nonlinear Hamiltonian systems with symmetry, Comp. Meth. Appl. Mech. Eng. 134:197–222 (1996).Google Scholar
  11. 11.
    I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, New York, 1965).Google Scholar
  12. 12.
    H. Grubmüller and P. Tavan, Molecular dynamics of conformational substates for a simplified protein model, J. Chem. Phys. 101:5047–5057 (1994).Google Scholar
  13. 13.
    J. Honerkamp, Stochastic Dynamical Systems: Concepts Numerical Methods, Data Analysis (VCH Publishers, New York, 1994).Google Scholar
  14. 14.
    V. Jakšić and C.-A. Pillet, Ergodic properties of the Langevin equation, Lett. Math. Phys. 41:49–57 (1997).Google Scholar
  15. 15.
    N. V. Krylov, Introduction to the Theory of Diffusion Processes, AMS Translations of Monographs, Volume 142 (1994).Google Scholar
  16. 16.
    C. Lubich, Integration of stiff mechanical systems by Runge-Kutta methods, ZAMP 44:1022–1053 (1993).Google Scholar
  17. 17.
    M. Mandziuk and T. Schlick, Resonance in the dynamics of chemical systems simulated by the implicit midpoint scheme, Chem. Phys. Lett. 237:525–535 (1995).Google Scholar
  18. 18.
    H. Mori, Transport, collective motion, and Brownian motion, Prog. Theor. Phys. 33:423–455 (1964).Google Scholar
  19. 19.
    S. Nordholm and R. Zwanzig, A systematic derivation of generalized Brownian motion theory, J. Stat. Phys. 13:347–371 (1975).Google Scholar
  20. 20.
    C. S. Peskin and T. Schlick, Molecular dynamics by the backward Euler method, Comm. Pure Appl. Math. XLII:1001–1031 (1989).Google Scholar
  21. 21.
    H. Rubin and P. Ungar, Motion under a strong constraining force, Comm. Pure. Appl. Math Appl. Math. X:65–87 (1957).Google Scholar
  22. 22.
    T. Schlick, M. Mandziuk, R. Skeel and K. Srinivas, Nonlinear resonance artifacts in molecular dynamics simulations, J. Comp. Phys. 140:1–29 (1998).Google Scholar
  23. 23.
    R. D. Skeel, G. Zhang, and T. Schlick, A family of symplectic integrators: Stability, accuracy and molecular dynamics applications, SIAM J. Sci. Comp. 18:203–222 (1997).Google Scholar
  24. 24.
    R. Zwanzig, Ensemble method in the theory of irreversibility, J. Chem. Phys. 33:1339–1341 (1960).Google Scholar

Copyright information

© Plenum Publishing Corporation 1999

Authors and Affiliations

  • A. M. Stuart
    • 1
    • 2
  • J. O. Warren
    • 3
    • 2
  1. 1.Mathematics Institute, Warwick UniversityCoventryEngland
  2. 2.Part of this work was performed while the authors were visitors at the Oxford University Computing LaboratoryEngland
  3. 3.Scientific Computing and Computational Mathematics ProgramStanford

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