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A new method of deriving reduced models of Hamiltonian dynamical systems isdeveloped using techniques from optimization and statistical estimation. Givena set of resolved variables that define a model reduction, thequasi-equilibrium ensembles associated with the resolved variables are employedas a family of trial probability densities on phase space. The residual thatresults from submitting these trial densities to the Liouville equation isquantified by an ensemble-averaged cost function related to the informationloss rate of the reduction. From an initial nonequilibrium state, thestatistical state of the system at any later time is estimated by minimizingthe time integral of the cost function over paths of trial densities.Statistical closure of the underresolved dynamics is obtained at the level ofthe value function, which equals the optimal cost of reduction with respect tothe resolved variables, and the evolution of the estimated statistical state isdeduced from the Hamilton-Jacobi equation satisfied by the value function. Inthe near-equilibrium regime, or under a local quadratic approximation in thefar-from-equilibrium regime, this best-fit closure is governed by adifferential equation for the estimated state vector coupled to a Riccatidifferential equation for the Hessian matrix of the value function. Sincememory effects are not explicitly included in the trial densities, a singleadjustable parameter is introduced into the cost function to capture atime-scale ratio between resolved and unresolved motions. Apart from thisparameter, the closed equations for the resolved variables are completelydetermined by the underlying deterministic dynamics.

Best-fit quasi-equilibrium ensembles: a general approach to statistical closure of underresolved Hamiltonian dynamics