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

The generalized canonical ensemble and its universal equivalence with the microcanonical ensemble

This paper shows for a general class of statistical mechanical models that when the microcanonical and canonical ensembles are nonequivalent on a subset of values of the energy, there often exists a generalized canonical ensemble that satisfies a strong form of equivalence with the microcanonical ensemble that we call universal equivalence. The generalized canonical ensemble that we consider is obtained from the standard canonical ensemble by adding an exponential factor involving a continuous function g of the Hamiltonian. For example, if the microcanonical entropy is C2, then universal equivalence of ensembles holds with g taken from a class of quadratic functions, giving rise to a generalized canonical ensemble known in the literature as the Gaussian ensemble. This use of functions g to obtain ensemble equivalence is a counterpart to the use of penalty functions and augmented Lagrangians in global optimization. linebreak Generalizing the paper by Ellis et al. [J. Stat. Phys. 101:999–1064 (2000)], we analyze the equivalence of the microcanonical and generalized canonical ensembles both at the level of equilibrium macrostates and at the thermodynamic level. A neat but not quite precise statement of one of our main results is that the microcanonical and generalized canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the generalized microcanonical entropy s–g is concave. This generalizes the work of Ellis et al., who basically proved that the microcanonical and canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the microcanonical entropy s is concave

Bostonia. Volume 14

Founded in 1900, Bostonia magazine is Boston University's main alumni publication, which covers alumni and student life, as well as university activities, events, and programs

A crossover study of short burst oxygen therapy (SBOT) for the relief of exercise-induced breathlessness in severe COPD

<p>Abstract</p> <p>Background</p> <p>Previous small studies suggested SBOT may be ineffective in relieving breathlessness after exercise in COPD.</p> <p>Methods</p> <p>34 COPD patients with FEV1 <40% predicted and resting oxygen saturation ≥93% undertook an exercise step test 4 times. After exercise, patients were given 4 l/min of oxygen from a simple face mask, 4 l/min air from a face mask (single blind), air from a fan or no intervention.</p> <p>Results</p> <p>Average oxygen saturation fell from 95.0% to 91.3% after exercise. The mean time to subjective recovery was 3.3 minutes with no difference between treatments. The mean Borg breathlessness score was 1.5/10 at rest, rising to 5.1/10 at the end of exercise (No breathlessness = 0, worst possible breathlessness = 10). Oxygen therapy had no discernable effect on Borg scores even for 14 patients who desaturated below 90%. 15 patients had no preferred treatment, 7 preferred oxygen, 6 preferred the fan, 3 preferred air via a mask and 3 preferred room air.</p> <p>Conclusions</p> <p>This study provides no support for the idea that COPD patients who are not hypoxaemic at rest derive noticeable benefit from oxygen therapy after exercise. Use of air from a mask or from a fan had no apparent physiological or placebo effect.</p

Generalized canonical ensembles and ensemble equivalence

This paper is a companion article to our previous paper (J. Stat. Phys. 119, 1283 (2005), cond-mat/0408681), which introduced a generalized canonical ensemble obtained by multiplying the usual Boltzmann weight factor $e^{-\beta H}$ of the canonical ensemble with an exponential factor involving a continuous function $g$ of the Hamiltonian $H$. We provide here a simplified introduction to our previous work, focusing now on a number of physical rather than mathematical aspects of the generalized canonical ensemble. The main result discussed is that, for suitable choices of $g$, the generalized canonical ensemble reproduces, in the thermodynamic limit, all the microcanonical equilibrium properties of the many-body system represented by $H$ even if this system has a nonconcave microcanonical entropy function. This is something that in general the standard ($g=0$) canonical ensemble cannot achieve. Thus a virtue of the generalized canonical ensemble is that it can be made equivalent to the microcanonical ensemble in cases where the canonical ensemble cannot. The case of quadratic $g$-functions is discussed in detail; it leads to the so-called Gaussian ensemble.Comment: 8 pages, 4 figures (best viewed in ps), revtex4. Changes in v2: Title changed, references updated, new paragraph added, minor differences with published versio

On variational principles for coherent vortex structures

Different approaches are discussed of variational principles characterizing coherent vortex structures in two-dimensional flows. Turbulent flows seem to form ordered structures in the large scales of the motion and the self-organization principle predicts asymptotic states realizing an extremal value of the energy or a minimum of enstrophy. On the other hand the small scales take care of the increase of entropy, and asymptotic results can be obtained by applying the theory of equilibrium statistical mechanics

Statistics of an Unstable Barotropic Jet from a Cumulant Expansion

Low-order equal-time statistics of a barotropic flow on a rotating sphere are investigated. The flow is driven by linear relaxation toward an unstable zonal jet. For relatively short relaxation times, the flow is dominated by critical-layer waves. For sufficiently long relaxation times, the flow is turbulent. Statistics obtained from a second-order cumulant expansion are compared to those accumulated in direct numerical simulations, revealing the strengths and limitations of the expansion for different relaxation times.Comment: 23 pages, 8 figures. Version to appear in J. Atmos. Sc

Rotational diversity effects in a triticale-based cropping system

Research indicates that not all crops respond similarly to cropping diversity and the response of triticale (× Triticosecale ssp.) has not been documented. We investigated the effects of rotational diversity on cereals in cropping sequences with canola (Brassica napus L.), field pea (Pisum sativum L.), or an intercrop (triticale:field pea). Six crop rotations were established consisting of two, 2-yr low diversity rotations (LDR) (continuous triticale (T-T_LDR) and triticale-wheat (Triticum aestivum L.) (T-W_LDR)); three, 2-yr moderate diversity rotations (MDR) (triticale-field pea (T-P_MDR), triticale-canola (T-C_MDR), and a triticale: field pea intercrop (T- in P_MDR)); and one, 3-yr high diversity rotation (HDR) (canola-triticale-field pea (C-T-P_HDR)). The study was established in Lethbridge, Alberta (irrigated and rainfed); Swift Current (rainfed) and Canora (rainfed), Saskatchewan, Canada; and carried out from 2008 to 2014. Triticale grain yield for the 3-yr HDR was superior over the LDR rotations and the MDR triticale-field pea system; however, results were similar for triticale-canola, and removal of canola from the system caused a yield drag in triticale. Triticale biomass was superior for the 3-yr HDR. Moreover, along with improved triticale grain yield, the 3-yr HDR provided greater yield stability across environments. High rotational diversity (C-T-P_HDR) resulted in the highest soil microbial community and soil carbon concentration, whereas continuous triticale provided the lowest. Net economic returns were also superior for C-T-P_HDR ($670 ha–1) and the lowest for T-W_LDR ($458 ha–1). Overall, triticale responded positively to increased rotational diversity and displayed greater stability with the inclusion of field pea, leading to improved profitability and sustainability of the system

Global Bifurcation of Rotating Vortex Patches

We rigorously construct continuous curves of rotating vortex patch solutions to the two-dimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum along the interface of the angular fluid velocity in the rotating frame becomes arbitrarily small. This is consistent with the conjectured existence [30, 38] of singular limiting patches with 90 corners at which the relative fluid velocity vanishes. For solutions close to the disk, we prove that there are “cat's-eyes”-type structures in the flow, and provide numerical evidence that these structures persist along the entire solution curves and are related to the formation of corners. We also show, for any rotating vortex patch, that the boundary is analytic as soon as it is sufficiently regular.</p