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Fitting SDE models to nonlinear Kac–Zwanzig heat bath models

Kupferman, R. and Stuart, A. M. (2004) Fitting SDE models to nonlinear Kac–Zwanzig heat bath models. Physica D, 199 (3-4). pp. 279-316. ISSN 0167-2789. doi:10.1016/j.physd.2004.04.011.

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We study a class of “particle in a heat bath” models, which are a generalization of the well-known Kac–Zwanzig class of models, but where the coupling between the distinguished particle and the n heat bath particles is through nonlinear springs. The heat bath particles have random initial data drawn from an equilibrium Gibbs density. The primary objective is to approximate the forces exerted by the heat bath—which we do not want to resolve—by a stochastic process. By means of the central limit theorem for Gaussian processes, and heuristics based on linear response theory, we demonstrate conditions under which it is natural to expect that the trajectories of the distinguished particle can be weakly approximated, as n→∞, by the solution of a Markovian SDE. The quality of this approximation is verified by numerical calculations with parameters chosen according to the linear response theory. Alternatively, the parameters of the effective equation can be chosen using time series analysis. This is done and agreement with linear response theory is shown to be good.

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Additional Information:© 2004 Elsevier B.V. Received 18 December 2002, Revised 28 October 2003, Accepted 6 April 2004, Available online 18 November 2004. We are grateful to D. Brillinger, A. Chorin, S. Evans, Y. Farjoun, R. Mannella, J. Neu, Y. Peres, P. Tupper and P. Wiberg for helpful discussions. RK was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities, by the Alon Fellowship, and by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the US Department of Energy under contract DE-AC03-76-SF00098. A.M.S. was supported by the EPSRC, UK.
Funding AgencyGrant Number
Israel Science FoundationUNSPECIFIED
Alon FellowshipUNSPECIFIED
Department of Energy (DOE)DE-AC03-76-SF00098
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
Subject Keywords:Hamiltonian systems; Heat bath; Stochastic differential equations; Nonlinear oscillators
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Other Numbering System NameOther Numbering System ID
Andrew StuartJ62
Issue or Number:3-4
Record Number:CaltechAUTHORS:20170609-143006730
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Official Citation:R. Kupferman, A.M. Stuart, Fitting SDE models to nonlinear Kac–Zwanzig heat bath models, Physica D: Nonlinear Phenomena, Volume 199, Issue 3, 2004, Pages 279-316, ISSN 0167-2789, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78067
Deposited By: Tony Diaz
Deposited On:09 Jun 2017 22:23
Last Modified:15 Nov 2021 17:36

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