Analysis of phase transitions in the mean-field Blume-Emery-Griffiths model

In this paper we give a complete analysis of the phase transitions in the mean-field Blume-Emery-Griffiths lattice-spin model with respect to the canonical ensemble, showing both a second-order, continuous phase transition and a first-order, discontinuous phase transition for appropriate values of the thermodynamic parameters that define the model. These phase transitions are analyzed both in terms of the empirical measure and the spin per site by studying bifurcation phenomena of the corresponding sets of canonical equilibrium macrostates, which are defined via large deviation principles. Analogous phase transitions with respect to the microcanonical ensemble are also studied via a combination of rigorous analysis and numerical calculations. Finally, probabilistic limit theorems for appropriately scaled values of the total spin are proved with respect to the canonical ensemble. These limit theorems include both central-limit-type theorems, when the thermodynamic parameters are not equal to critical values, and noncentral-limit-type theorems, when these parameters equal critical values.Comment: Published at http://dx.doi.org/10.1214/105051605000000421 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic behavior of the finite-size magnetization as a function of the speed of approach to criticality

The main focus of this paper is to determine whether the thermodynamic magnetization is a physically relevant estimator of the finite-size magnetization. This is done by comparing the asymptotic behaviors of these two quantities along parameter sequences converging to either a second-order point or the tricritical point in the mean-field Blume--Capel model. We show that the thermodynamic magnetization and the finite-size magnetization are asymptotic when the parameter $\alpha$ governing the speed at which the sequence approaches criticality is below a certain threshold $\alpha_0$. However, when $\alpha$ exceeds $\alpha_0$, the thermodynamic magnetization converges to 0 much faster than the finite-size magnetization. The asymptotic behavior of the finite-size magnetization is proved via a moderate deviation principle when $0\alpha_0$. To the best of our knowledge, our results are the first rigorous confirmation of the statistical mechanical theory of finite-size scaling for a mean-field model.Comment: Published in at http://dx.doi.org/10.1214/10-AAP679 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Ginzburg-Landau Polynomials and the Asymptotic Behavior of the Magnetization Near Critical and Tricritical Points

For the mean-field version of an important lattice-spin model due to Blume and Capel, we prove unexpected connections among the asymptotic behavior of the magnetization, the structure of the phase transitions, and a class of polynomials that we call the Ginzburg-Landau polynomials. The model depends on the parameters n, beta, and K, which represent, respectively, the number of spins, the inverse temperature, and the interaction strength. Our main focus is on the asymptotic behavior of the magnetization m(beta_n,K_n) for appropriate sequences (beta_n,K_n) that converge to a second-order point or to the tricritical point of the model and that lie inside various subsets of the phase-coexistence region. The main result states that as (beta_n,K_n) converges to one of these points (beta,K), m(beta_n,K_n) ~ c |beta - beta_n|^gamma --> 0. In this formula gamma is a positive constant, and c is the unique positive, global minimum point of a certain polynomial g that we call the Ginzburg-Landau polynomial. This polynomial arises as a limit of appropriately scaled free-energy functionals, the global minimum points of which define the phase-transition structure of the model. For each sequence (beta_n,K_n) under study, the structure of the global minimum points of the associated Ginzburg-Landau polynomial mirrors the structure of the global minimum points of the free-energy functional in the region through which (beta_n,K_n) passes and thus reflects the phase-transition structure of the model in that region. The properties of the Ginzburg-Landau polynomials make rigorous the predictions of the Ginzburg-Landau phenomenology of critical phenomena, and the asymptotic formula for m(beta_n,K_n) makes rigorous the heuristic scaling theory of the tricritical point.Comment: 70 pages, 8 figure

On Shear-Free perturbations of FLRW Universes

A surprising exact result for the Einstein Field Equations is that if pressure-free matter is moving in a shear-free way, then it must be either expansion-free or rotation-free. It has been suggested this result is also true for any barotropic perfect fluid, but a proof has remained elusive. We consider the case of barotropic perfect fluid solutions linearized about a Robertson-Walker geometry, and prove that the result remains true except for the case of a specific highly non-linear equation of state. We argue that this equation of state is non-physical, and hence the result is true in the linearized case for all physically realistic barotropic perfect fluids. This result, which is not true in Newtonian cosmology, demonstrates that the linearized solutions, believed to result in standard local Newtonian theory, do not always give the usual behaviour of Newtonian solutions

Numerical evaluation of one-loop QCD amplitudes

We present the publicly available program NGluon allowing the numerical evaluation of primitive amplitudes at one-loop order in massless QCD. The program allows the computation of one-loop amplitudes for an arbitrary number of gluons. The focus of the present article is the extension to one-loop amplitudes including an arbitrary number of massless quark pairs. We discuss in detail the algorithmic differences to the pure gluonic case and present cross checks to validate our implementation. The numerical accuracy is investigated in detail.Comment: Talk given at ACAT 2011 conference in London, 5-9 Septembe

Weak lensing B-modes on all scales as a probe of local isotropy

This article derives a multipolar hierarchy for the propagation of the weak-lensing shear and convergence in a general spacetime. The origin of B-modes, in particular on large angular scales, is related to the local isotropy of space. Known results assuming a Friedmann-Lema\^itre background are naturally recovered. The example of a Bianchi I spacetime illustrates our formalism and its implications for future observations are stressed.Comment: 10 pages, 2 figures. Replaced to match published versio

A Geometrical Approach to Strong Gravitational Lensing in f(R) Gravity

We present a framework for the study of lensing in spherically symmetric spacetimes within the context of f(R) gravity. Equations for the propagation of null geodesics, together with an expression for the bending angle are derived for any f(R) theory and then applied to an exact spherically symmetric solution of R^n gravity. We find that for this case more bending is expected for R^n gravity theories in comparison to GR and is dependent on the value of n and the value of distance of closest approach of the incident null geodesic.Comment: 9 page

A genetic basis for a postmeiotic X versus Y chromosome intragenomic conflict in the mouse.

Intragenomic conflicts arise when a genetic element favours its own transmission to the detriment of others. Conflicts over sex chromosome transmission are expected to have influenced genome structure, gene regulation, and speciation. In the mouse, the existence of an intragenomic conflict between X- and Y-linked multicopy genes has long been suggested but never demonstrated. The Y-encoded multicopy gene Sly has been shown to have a predominant role in the epigenetic repression of post meiotic sex chromatin (PMSC) and, as such, represses X and Y genes, among which are its X-linked homologs Slx and Slxl1. Here, we produced mice that are deficient for both Sly and Slx/Slxl1 and observed that Slx/Slxl1 has an opposite role to that of Sly, in that it stimulates XY gene expression in spermatids. Slx/Slxl1 deficiency rescues the sperm differentiation defects and near sterility caused by Sly deficiency and vice versa. Slx/Slxl1 deficiency also causes a sex ratio distortion towards the production of male offspring that is corrected by Sly deficiency. All in all, our data show that Slx/Slxl1 and Sly have antagonistic effects during sperm differentiation and are involved in a postmeiotic intragenomic conflict that causes segregation distortion and male sterility. This is undoubtedly what drove the massive gene amplification on the mouse X and Y chromosomes. It may also be at the basis of cases of F1 male hybrid sterility where the balance between Slx/Slxl1 and Sly copy number, and therefore expression, is disrupted. To the best of our knowledge, our work is the first demonstration of a competition occurring between X and Y related genes in mammals. It also provides a biological basis for the concept that intragenomic conflict is an important evolutionary force which impacts on gene expression, genome structure, and speciation

The Existence of Einstein Static Universes and their Stability in Fourth order Theories of Gravity

We investigate whether or not an Einstein Static universe is a solution to the cosmological equations in $f(R)$ gravity. It is found that only one class of $f(R)$ theories admits an Einstein Static model, and that this class is neutrally stable with respect to vector and tensor perturbations for all equations of state on all scales. Scalar perturbations are only stable on all scales if the matter fluid equation of state satisfies $c_s^2>\frac{\sqrt{5}-1}{6}\approx 0.21$. This result is remarkably similar to the GR case, where it was found that the Einstein Static model is stable for $c_s^2>{1/5}$.Comment: Minor changes, To appear in PR