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. ]]>

arise from the coarse-graining of a random field. Coarse-grained processes of this

kind form the basis of the analysis of local mean-field models in statistical mechanics

by exploiting the long-range nature of the interaction function defining such models.

In particular, the large deviation principle is used in a companion paper [8] to

derive the variational principles that characterize equilibrium macrostates in statistical

models of two-dimensional and quasi-geostrophic turbulence. Such macrostates

correspond to large-scale, long-lived flow structures, the description of which is the

goal of the statistical equilibrium theory of turbulence. The large deviation bounds

for the coarse-grained process under consideration are shown to hold with respect

to the strong L2 topology, while the associated rate function is proved to have compact

level sets with respect to the weak topology. This compactness property is

nevertheless sufficient to establish the existence of equilibrium macrostates for both

the microcanonical and canonical ensembles. ]]>

problem having the same minimum points? This basic question in global optimization

motivates this paper, which answers it from the viewpoint of statistical mechanics. In this context, it

reduces to the fundamental question of the equivalence and nonequivalence of ensembles, which is

analyzed using the theory of large deviations and the theory of convex functions.

In a 2000 paper appearing in the Journal of Statistical Physics, we gave necessary and sufficient

conditions for ensemble equivalence and nonequivalence in terms of support and concavity

properties of the microcanonical entropy. In later research we significantly extended those results

by introducing a class of Gaussian ensembles, which are obtained from the canonical ensemble by

adding an exponential factor involving a quadratic function of the Hamiltonian. The present paper

is an overview of our work on this topic. Our most important discovery is that even when the microcanonical

and canonical ensembles are not equivalent, one can often find a Gaussian ensemble

that satisfies a strong form of equivalence with the microcanonical ensemble known as universal

equivalence. When translated back into optimization theory, this implies that an unconstrained minimization

problem involving a Lagrange multiplier and a quadratic penalty function has the same

minimum points as the original constrained problem.

The results on ensemble equivalence discussed in this paper are illustrated in the context of the

Curie-Weiss-Potts lattice-spin model. ]]>