#### Publication Date

2010

#### Abstract

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.

#### Recommended Citation

Turkington, B and Plechac, P, "Best-fit quasi-equilibrium ensembles: a general approach to statistical closure of underresolved Hamiltonian dynamics" (2010). *Mathematics and Statistics Department Faculty Publication Series*. 1206.

Retrieved from https://scholarworks.umass.edu/math_faculty_pubs/1206

## Comments

This is the prepublished version harvested from ArXiv.