Mathematics and Statistics Department Faculty Publication Series

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

Turkington, B et al. — Mon, 09 May 2011 13:29:59 PDT

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.

The Large Deviation Principle for Coarse-Grained Processes

Ellis, RS et al. — Mon, 09 May 2011 13:29:47 PDT

The large deviation principle is proved for a class of L2-valued processes that
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.

Group actions on 4-manifolds: some recent results and open questions

Chen, WM — Thu, 31 Mar 2011 12:03:06 PDT

A survey of finite group actions on symplectic 4-manifolds is given with a special emphasis on results and questions concerning smooth or symplectic classification of group actions, group actions and exotic smooth structures, and homological rigidity and boundedness of group actions. We also take this opportunity to include several results and questions which did not appear elsewhere.

ON A NOTION OF MAPS BETWEEN ORBIFOLDS II: HOMOTOPY AND CW-COMPLEX

Chen, WM — Thu, 31 Mar 2011 12:02:54 PDT

This is the second of a series of papers which is devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants — the homotopy groups, and (2) an analog of CW-complex theory. As a corollary of this machinery, the classical Whitehead theorem (which asserts that a weak homotopy equivalence is a homotopy equivalence) is extended to the orbifold category.

On the orders of periodic diffeomorphisms of 4-manifolds

Chen, WM — Thu, 31 Mar 2011 12:02:41 PDT

This paper initiated an investigation on the following question: Suppose that a smooth 4 -manifold does not admit any smooth circle actions. Does there exist a constant C>0 such that the manifold supports no smooth Zp -actions of prime order for p>C ? We gave affirmative results to this question for the case of holomorphic and symplectic actions, with an interesting finding that the constant C in the holomorphic case is topological in nature, while in the symplectic case it involves also the smooth structure of the manifold.

Symmetric symplectic homotopy K3 surfaces

Chen, WM et al. — Thu, 31 Mar 2011 12:02:29 PDT

A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is shown that an effective action by various maximal symplectic K3 groups forces the corresponding homotopy K3 surface to be minimally exotic with respect to our measure. (However, the standard K3 is the only known example of such minimally exotic homotopy K3 surfaces.) The possible structure of a finite group of symplectic symmetries of a minimally exotic homotopy K3 surface is determined and future research directions are indicated.

KIDA’S FORMULA AND CONGRUENCES

Pollack, R et al. — Wed, 30 Mar 2011 13:34:23 PDT

ON ANTICYCLOTOMIC ì-INVARIANTS OF MODULAR FORMS

Pollack, R et al. — Wed, 30 Mar 2011 13:34:04 PDT

Global Optimization, the Gaussian Ensemble, and Universal Ensemble Equivalence.

Costeniuc, M et al. — Wed, 30 Mar 2011 12:55:28 PDT

Given a constrained minimization problem, under what conditions does there exist a related, unconstrained
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.

Nonequivalent ensembles and metastability

TOUCHETT, H et al. — Wed, 30 Mar 2011 12:55:11 PDT

This paper reviews a number of fundamental connections that exist between nonequivalent microcanonical and canonical ensembles, the appearance of first-order phase transitions in the canonical ensemble, and thermodynamic metastable behavior.