A new method of deriving reduced models of Hamiltonian dynamical systems is
developed using techniques from optimization and statistical estimation. Given
a set of resolved variables that define a model reduction, the
quasi-equilibrium ensembles associated with the resolved variables are employed
as a family of trial probability densities on phase space. The residual that
results from submitting these trial densities to the Liouville equation is
quantified by an ensemble-averaged cost function related to the information
loss rate of the reduction. From an initial nonequilibrium state, the
statistical state of the system at any later time is estimated by minimizing
the time integral of the cost function over paths of trial densities.
Statistical closure of the underresolved dynamics is obtained at the level of
the value function, which equals the optimal cost of reduction with respect to
the resolved variables, and the evolution of the estimated statistical state is
deduced from the Hamilton-Jacobi equation satisfied by the value function. In
the near-equilibrium regime, or under a local quadratic approximation in the
far-from-equilibrium regime, this best-fit closure is governed by a
differential equation for the estimated state vector coupled to a Riccati
differential equation for the Hessian matrix of the value function. Since
memory effects are not explicitly included in the trial densities, a single
adjustable parameter is introduced into the cost function to capture a
time-scale ratio between resolved and unresolved motions. Apart from this
parameter, the closed equations for the resolved variables are completely
determined by the underlying deterministic dynamics
