Journal of Statistical Physics

, Volume 97, Issue 3–4, pp 687–723 | Cite as

Analysis and Experiments for a Computational Model of a Heat Bath

  • A. M. Stuart
  • J. O. Warren


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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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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