TY - JOUR

T1 - A Nonequilibrium Statistical Model of Spectrally Truncated Burgers-Hopf Dynamics

AU - Kleeman, Richard

AU - Turkington, Bruce E.

N1 - Funding Information:
Eva Johansson and Per Lindberg, Department of Public Health and Caring Sciences, Section of Caring Sciences, Uppsala Science Park, Uppsala, Sweden. This study was supported by the Swedish Foundation for Health Care Sciences and Allergy Research. Correspondence concerning this article should be addressed to Eva Johansson, Department of Public Health and Caring Sciences, Section of Caring Sciences, Uppsala Science Park, S–751 83 Uppsala, Sweden. E-mail: eva.johansson@ccs.uu.se
Publisher Copyright:
© 2014 Wiley Periodicals, Inc.

PY - 2014/12/1

Y1 - 2014/12/1

N2 - Exact spectral truncations of the unforced, inviscid Burgers-Hopf equation are Hamiltonian systems with many degrees of freedom that exhibit intrinsic stochasticity and coherent scaling behavior. For this reason recent studies have employed these systems as prototypes to test stochastic mode reduction strategies. In the present paper the Burgers-Hopf dynamics truncated to n Fourier modes is treated by a new statistical model reduction technique, and a closed system of evolution equations for the mean values of the m lowest modes is derived for m≪n. In the reduced model the m-mode macrostates are associated with trial probability densities on the phase space of the n-mode microstates, and a cost functional is introduced to quantify the lack of fit of paths of these densities to the Liouville equation. The best-fit macrodynamics is obtained by minimizing the cost functional over paths, and the equations governing the closure are then derived from Hamilton-Jacobi theory. The resulting reduced equations have a fractional diffusion and modified nonlinear interactions, and the explicit form of both are determined up to a single closure parameter. The accuracy and range of validity of this nonequilibrium closure is assessed by comparison against direct numerical simulations of statistical ensembles, and the predicted behavior is found to be well represented by the reduced equations.

AB - Exact spectral truncations of the unforced, inviscid Burgers-Hopf equation are Hamiltonian systems with many degrees of freedom that exhibit intrinsic stochasticity and coherent scaling behavior. For this reason recent studies have employed these systems as prototypes to test stochastic mode reduction strategies. In the present paper the Burgers-Hopf dynamics truncated to n Fourier modes is treated by a new statistical model reduction technique, and a closed system of evolution equations for the mean values of the m lowest modes is derived for m≪n. In the reduced model the m-mode macrostates are associated with trial probability densities on the phase space of the n-mode microstates, and a cost functional is introduced to quantify the lack of fit of paths of these densities to the Liouville equation. The best-fit macrodynamics is obtained by minimizing the cost functional over paths, and the equations governing the closure are then derived from Hamilton-Jacobi theory. The resulting reduced equations have a fractional diffusion and modified nonlinear interactions, and the explicit form of both are determined up to a single closure parameter. The accuracy and range of validity of this nonequilibrium closure is assessed by comparison against direct numerical simulations of statistical ensembles, and the predicted behavior is found to be well represented by the reduced equations.

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U2 - 10.1002/cpa.21498

DO - 10.1002/cpa.21498

M3 - Article

AN - SCOPUS:84908879498

VL - 67

SP - 1905

EP - 1946

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 12

ER -