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An Optimization Principle for Deriving Nonequilibrium Statistical Models of Hamiltonian Dynamics

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Abstract

A general method for deriving closed reduced models of Hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation. Given a vector of resolved variables, selected to describe the macroscopic state of the system, a family of quasi-equilibrium probability densities on phase space corresponding to the resolved variables is employed as a statistical model, and the evolution of the mean resolved vector is estimated by optimizing over paths of these densities. Specifically, a cost function is constructed to quantify the lack-of-fit to the microscopic dynamics of any feasible path of densities from the statistical model; it is an ensemble-averaged, weighted, squared-norm of the residual that results from submitting the path of densities to the Liouville equation. The path that minimizes the time integral of the cost function determines the best-fit evolution of the mean resolved vector. The closed reduced equations satisfied by the optimal path are derived by Hamilton-Jacobi theory. When expressed in terms of the macroscopic variables, these equations have the generic structure of governing equations for nonequilibrium thermodynamics. In particular, the value function for the optimization principle coincides with the dissipation potential that defines the relation between thermodynamic forces and fluxes. The adjustable closure parameters in the best-fit reduced equations depend explicitly on the arbitrary weights that enter into the lack-of-fit cost function. Two particular model reductions are outlined to illustrate the general method. In each example the set of weights in the optimization principle contracts into a single effective closure parameter.

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Acknowledgements

In the course of this work the author benefited from conversations with R.S. Ellis, M. Katsoulakis, R. Kleeman, A.J. Majda, and P. Plechac. This research was initiated during a sabbatical stay at the University of Warwick partly supported by an international short visit fellowship from the Royal Society, and was completed during a two-month visit to the Courant Institute of Mathematical Sciences. This work received funding from the National Science Foundation under grant DMS-0604071.

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  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003, USA

    Bruce Turkington

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Turkington, B. An Optimization Principle for Deriving Nonequilibrium Statistical Models of Hamiltonian Dynamics. J Stat Phys 152, 569–597 (2013). https://doi.org/10.1007/s10955-013-0778-9

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