From a general and theoretical standpoint, subgrid-scale parameterization can be viewed as a problem in nonequilibrium statistical mechanics. Given a microscopic dynamics that is high-dimensional and chaotic, one seeks to determine a closed set of equations for some resolved variables whose expectations are the macroscopic observables of the system. I will present a new methodology for statistical model reduction of complex deterministic systems from this perspective. The key idea is to choose a convenient family of probability densities on phase space corresponding to the resolved macrostates, and to quantify their residual with respect to the Liouville equation. By minimizing the Liouville residual over the time interval of evolution, the best-fit densities define a statistical closure, whose macroscopic governing equations have the form of irreversible thermodynamics. I will illustrate this method for the near-equilibrium relaxation of Hamiltonian systems. The ultimate goal of this work is to provide a systematic procedure for devising and diagnosing closure schemes for turbulent dynamics.
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