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

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

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

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

Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution

Using a Maximum Entropy Production Principle (MEPP), we derive a new type of relaxation equations for two-dimensional turbulent flows in the case where a prior vorticity distribution is prescribed instead of the Casimir constraints [Ellis, Haven, Turkington, Nonlin., 15, 239 (2002)]. The particular case of a Gaussian prior is specifically treated in connection to minimum enstrophy states and Fofonoff flows. These relaxation equations are compared with other relaxation equations proposed by Robert and Sommeria [Phys. Rev. Lett. 69, 2776 (1992)] and Chavanis [Physica D, 237, 1998 (2008)]. They can provide a small-scale parametrization of 2D turbulence or serve as numerical algorithms to compute maximum entropy states with appropriate constraints. We perform numerical simulations of these relaxation equations in order to illustrate geometry induced phase transitions in geophysical flows.Comment: 21 pages, 9 figure

A new positive relationship between pCO2 and stomatal frequency in Quercus guyavifolia (Fagaceae): a potential proxy for palaeo-CO2 levels

Background and Aims The inverse relationship between atmospheric CO2 partial pressure (pCO2) and stomatal frequency in many species of plants has been widely used to estimate palaeoatmospheric CO2 (palaeo-CO2) levels; however, the results obtained have been quite variable. This study attempts to find a potential new proxy for palaeo-CO2 levels by analysing stomatal frequency in Quercus guyavifolia (Q. guajavifolia, Fagaceae), an extant dominant species of sclerophyllous forests in the Himalayas with abundant fossil relatives. Methods Stomatal frequency was analysed for extant samples of Q. guyavifolia collected from17 field sites at altitudes ranging between 2493 and 4497 m. Herbarium specimens collected between 1926 and 2011 were also examined. Correlations of pCO2-stomatal frequency were determined using samples from both sources, and these were then applied to Q. preguyavaefolia fossils in order to estimate palaeo-CO2 concentrations for two late-Pliocene floras in south-western China. Key Results In contrast to the negative correlations detected for most other species that have been studied, a positive correlation between pCO2 and stomatal frequency was determined in Q. guyavifolia sampled from both extant field collections and historical herbarium specimens. Palaeo-CO2 concentrations were estimated to be approx. 180-240 ppm in the late Pliocene, which is consistent with most other previous estimates. Conclusions A new positive relationship between pCO2 and stomatal frequency in Q. guyavifolia is presented, which can be applied to the fossils closely related to this species that are widely distributed in the late-Cenozoic strata in order to estimate palaeo-CO2 concentrations. The results show that it is valid to use a positive relationship to estimate palaeo-CO2 concentrations, and the study adds to the variety of stomatal density/index relationships that available for estimating pCO2. The physiological mechanisms underlying this positive response are unclear, however, and require further researc

Constrained and unconstrained rearrangement minimization problems related to the p-Laplace operator

In this paper we consider an unconstrained and a constrained minimization problem related to the boundary value problem −∆pu = f in D, u = 0 on ∂D. In the unconstrained problem we minimize an energy functional relative to a rearrangement class, and prove existence of a unique solution. We also consider the case when D is a planar disk and show that the minimizer is radial and increasing. In the constrained problem we minimize the energy functional relative to the intersection of a rearrangement class with an affine subspace of codimension one in an appropriate function space. We briefly discuss our motivation for studying the constrained minimization problem

``Smoke Rings'' in Ferromagnets

It is shown that bulk ferromagnets support propagating non-linear modes that are analogous to the vortex rings, or ``smoke rings'', of fluid dynamics. These are circular loops of {\it magnetic} vorticity which travel at constant velocity parallel to their axis of symmetry. The topological structure of the continuum theory has important consequences for the properties of these magnetic vortex rings. One finds that there exists a sequence of magnetic vortex rings that are distinguished by a topological invariant (the Hopf invariant). We present analytical and numerical results for the energies, velocities and structures of propagating magnetic vortex rings in ferromagnetic materials.Comment: 4 pages, 3 eps-figures, revtex with epsf.tex and multicol.sty. To appear in Physical Review Letters. (Postscript problem fixed.

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