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

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

On The Differentiation Of A Log-Liklihood Function Using Matrix Calculus

Simple theorems based on a mathematical property of vecY/vecX provide powerful tools for obtaining matrix calculus results. By way of illustration, new results are obtained for matrix derivatives involving vecA, vechA, v(A) and vecX where X is a symmetric matrix. The analysis explains exactly how a log-likelihood function should be differentiated using matrix calculus.

A Mean-field statistical theory for the nonlinear Schrodinger equation

A statistical model of self-organization in a generic class of one-dimensional nonlinear Schrodinger (NLS) equations on a bounded interval is developed. The main prediction of this model is that the statistically preferred state for such equations consists of a deterministic coherent structure coupled with fine-scale, random fluctuations, or radiation. The model is derived from equilibrium statistical mechanics by using a mean-field approximation of the conserved Hamiltonian and particle number for finite-dimensional spectral truncations of the NLS dynamics. The continuum limits of these approximated statistical equilibrium ensembles on finite-dimensional phase spaces are analyzed, holding the energy and particle number at fixed, finite values. The analysis shows that the coherent structure minimizes total energy for a given value of particle number and hence is a solution to the NLS ground state equation, and that the remaining energy resides in Gaussian fluctuations equipartitioned over wavenumbers. Some results of direct numerical integration of the NLS equation are included to validate empirically these properties of the most probable states for the statistical model. Moreover, a theoretical justification of the mean-field approximation is given, in which the approximate ensembles are shown to concentrate on the associated microcanonical ensemble in the continuum limit.Comment: 24 pages, 2 figure

Large Deviation Principles and Complete Equivalence and Nonequivalence Results for Pure and Mixed Ensembles

We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of two-dimensional fluid motion, quasi-geostrophic flows, and dispersive waves. First, large deviation principles are proved for the canonical ensemble and the microcanonical ensemble. For each ensemble the set of equilibrium macrostates is defined as the set on which the corresponding rate function attains its minimum of 0. We then present complete equivalence and nonequivalence results at the level of equilibrium macrostates for the two ensembles.Comment: 57 page

The Large Deviation Principle for Coarse-Grained Processes

The large deviation principle is proved for a class of $L^2$-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 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 $L^2$ 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.Comment: 19 page

Linguistic analysis of the valence, arousal and dominance of auditory hallucinations and internal thoughts in schizophrenia: Implications for psychoeducation and CBT

70% of patients with schizophrenia suffer from auditory verbal hallucinations (AVH) which are frequently described as distressing and disabling. The content of AVH, in relation to internal thought, has never been linguistically tested in a self-monitoring study. The aim of this preliminary study was to establish if there was a significant difference between AVH and inner thoughts on the key linguistic parameters of valence (pleasantness), dominance (control) and arousal (intensity of emotion produced). Six volunteers with a diagnosis of schizophrenia from voice hearing support groups produced real-time, detailed diaries of AVH and inner thoughts using randomised/fixed timers. Analysis of content was completed using an established linguistic database. AVH were significantly more unpleasant and controlling but not more emotionally arousing than inner thoughts. Psychoeducation around the experience of hallucination in schizophrenia should include information that the voices will be significantly more unpleasant and controlling than their own thoughts but not more emotionally arousing. CBT might therefore include the use of compassion focussed techniques to help with the unpleasantness of AVH and schema level techniques to improve coping with the dominance of AVH