Expression For Divergence Research Articles

An Ising Model for Supercooled Liquids and the Glass Transition

We describe the behavior of an Ising model with orthogonal dynamics, where changes in energy and changes in alignment never occur during the same Monte Carlo (MC) step. This orthogonal Ising model (OIM) allows conservation of energy and conservation of (angular) momentum to proceed independently, on their own preferred time scales. The OIM also includes a third type of MC step that makes or breaks the interaction between neighboring spins, facilitating an equilibrium distribution of bond energies. MC simulations of the OIM mimic more than twenty distinctive characteristics that are commonly found above and below the glass temperature, Tg. Examples include a specific heat that has hysteresis around Tg, out-of-phase (loss) response that exhibits primary (α) and secondary (β) peaks, super-Arrhenius T dependence for the α-response time (τα), and fragilities that increase with increasing system size (N). Mean-field theory for energy fluctuations in the OIM yields a critical temperature (Tc) and a novel expression for the super-Arrhenius divergence as T→Tc: ln(τα)~1/(1−Tc/T)2. Because this divergence is reminiscent of the Vogel-Fulcher-Tammann (VFT) law squared, we call it the “VFT2 law”. A modified Stickel plot, which linearizes the VFT2 law, shows that at high T where mean-field theory should apply, only the VFT2 law gives qualitatively consistent agreement with measurements of τα (from the literature) on five glass-forming liquids. Such agreement with the OIM suggests that several basic features govern supercooled liquids. The freezing of a liquid into a glass involves an underlying 2nd-order transition that is broadened by finite-size effects. The VFT2 law for τα comes from energy fluctuations that enhance the pathways through an entropy bottleneck, not activation over an energy barrier. Values of τα vary exponentially with inverse N, consistent with the distribution of relaxation times deduced from measurements of α response. System sizes found via the T dependence of τα from simulations and measurements are similar to sizes of independently relaxing regions (IRR) measured by nuclear magnetic resonance (NMR) for simple-molecule glass-forming liquids. The OIM elucidates the key ingredients needed to interpret the thermal and dynamic properties of amorphous materials, while providing a broad foundation for more-detailed models of liquid-glass behavior.

Differential gene expression reveals mechanisms related to habitat divergence between hybridizing orchids from the Neotropical coastal plains

BackgroundClosely related hybridizing species are ideal systems for identifying genomic regions underlying adaptive divergence. Although gene expression plays a central role in determining ecologically-based phenotypic differences, few studies have inferred the role of gene expression for adaptive divergence in Neotropical systems. In this study, we conduct genome-wide expression analysis alongside soil elemental analysis in sympatric and allopatric populations of Epidendrum fulgens and E. puniceoluteum (Orchidaceae), which occur in contrasting adjacent habitats in the Neotropical coastal plains.ResultsThese species were highly differentiated by their gene expression profiles, as determined by 18–21% of transcripts. Gene ontology (GO) terms associated with reproductive processes were enriched according to comparisons between species in both allopatric and sympatric populations. Species showed differential expression in genes linked to salt and waterlogging tolerance according to comparisons between species in sympatry, and biological processes related to environmental stimulus appeared as representative among those transcripts associated with edaphic characteristics in each sympatric zone. Hybrids, in their turn, were well differentiated from E. fulgens, but exhibited a similar gene expression profile to flooding-tolerant E. puniceolutem. When compared with parental species, hybrids showed no transcripts with additive pattern of expression and increased expression for almost all transgressive transcripts.ConclusionsThis study sheds light on general mechanisms promoting ecological differentiation and assortative mating, and suggests candidate genes, such as those encoding catalase and calcium-dependent protein kinase, underling adaptation to harsh edaphic conditions in the Neotropical coastal plains. Moreover, it demonstrates that differential gene expression plays a central role in determining ecologically-based phenotypic differences among co-occurring species and their hybrids.

Vacuum birefringence in the head-on collision of x-ray free-electron laser and optical high-intensity laser pulses

The focus of this article is on providing compact analytical expressions for the differential number of polarization flipped signal photons constituting the signal of vacuum birefringence in the head-on collision of x-ray free electron (XFEL) and optical high-intensity laser pulses. Our results allow for unprecedented insights into the scaling of the effect with the waists and pulse durations of both laser beams, the Rayleigh range of the high-intensity beam, as well as transverse and longitudinal offsets. They account for the decay of the differential number of signal photons in the far-field as a function of the azimuthal angle measured relative to the beam axis of the probe beam in forward direction, typically neglected by conventional approximations. Moreover, they even allow us to extract an analytical expression for the angular divergence of the perpendicularly polarized signal photons. We expect our formulas to be very useful for the planning and optimization of experimental scenarios aiming at the detection of vacuum birefringence in XFEL/high-intensity laser setups, such as the one put forward at the Helmholtz International Beamline for Extreme Fields (HIBEF) at the European XFEL.

Minimising the Kullback-Leibler Divergence for Model Selection in Distributed Nonlinear Systems.

The Kullback–Leibler (KL) divergence is a fundamental measure of information geometry that is used in a variety of contexts in artificial intelligence. We show that, when system dynamics are given by distributed nonlinear systems, this measure can be decomposed as a function of two information-theoretic measures, transfer entropy and stochastic interaction. More specifically, these measures are applicable when selecting a candidate model for a distributed system, where individual subsystems are coupled via latent variables and observed through a filter. We represent this model as a directed acyclic graph (DAG) that characterises the unidirectional coupling between subsystems. Standard approaches to structure learning are not applicable in this framework due to the hidden variables; however, we can exploit the properties of certain dynamical systems to formulate exact methods based on differential topology. We approach the problem by using reconstruction theorems to derive an analytical expression for the KL divergence of a candidate DAG from the observed dataset. Using this result, we present a scoring function based on transfer entropy to be used as a subroutine in a structure learning algorithm. We then demonstrate its use in recovering the structure of coupled Lorenz and Rössler systems.

Five-Loop Four-Point Amplitude ofN=4Super-Yang-Mills Theory

Using the method of maximal cuts, we construct the complete D-dimensional integrand of the five-loop four-point amplitude of N=4 super-Yang-Mills theory, including nonplanar contributions. In the critical dimension where this amplitude becomes ultraviolet divergent, we present a compact explicit expression for the nonvanishing ultraviolet divergence in terms of three vacuum integrals. This construction provides a crucial step towards obtaining the corresponding amplitude of N=8 supergravity required to resolve the general ultraviolet behavior of supergravity theories.

Bhattacharyya statistical divergence of quantum observables

In this article we exploit the Bhattacharyya statistical divergence to determine the similarity of probability distributions of quantum observables. After brief review of useful characteristics of the Bhattacharyya divergence we apply it to determine the similarity of probability distributions of two noncommuting observables. An explicit expression for the Bhattacharyya statistical divergence is found for the case of two observables which are the x - and z -components of the angular momentum of a spin-1/2 system. Finally, a note is given of application of the considered statistical divergence to the specific physical measurement.

A robust, colocated, implicit algorithm for direct numerical simulation of compressible, turbulent flows

A non-dissipative, robust, implicit algorithm is proposed for direct numerical and large-eddy simulation of compressible turbulent flows. The algorithm addresses the problems caused by low Mach numbers and under-resolved high Reynolds numbers. It colocates variables in space to allow easy extension to unstructured grids, and discretely conserves mass, momentum and total energy. The Navier–Stokes equations are non-dimensionalized using an incompressible scaling for pressure, and the energy equation is used to obtain an expression for the velocity divergence. A pressure-correction approach is used to solve the resulting equations, such that the discrete divergence is constrained by the energy equation. As a result, the discrete equations analytically reduce to the incompressible equations at very low Mach number, i.e., the algorithm overcomes the acoustic time-scale limit without preconditioning or solution of an implicit system of equations. The algorithm discretely conserves kinetic energy in the incompressible inviscid limit, and is robust for inviscid compressible turbulence on the convective time-scale. These properties make it well-suited for DNS/LES of compressible turbulent flows. Results are shown for acoustic propagation, the incompressible Taylor problem, periodic shock tube problem, and isotropic turbulence.

The eigenray method for rapidly estimating refraction effects in a linear profile atmosphere

Eigenray solutions to sound propagation in an atmosphere with a linear sound-speed profile offer a convenient technique to estimate the effects of refraction. Eigenrays represent the trajectories of wave fronts that depart from the source and reach a given receiver. In a linear profile, the rays are arcs of circles, and over flat ground with perfect reflection, the paths can be constructed from a series of hops. Improvements to the eigenray technique are proposed that match the exact solution for pressure and phase in the unbounded linear profile. For reflection, the choice of a correct branch in the expression for divergence is necessary so that adjacent receivers with different numbers of eigenrays report the same pressure. Extension of the technique to lend estimates for some more complex situations is straightforward, although it fails to offer an advantage near caustics and focal regions.

Three-loop finiteness conditions in N = 1 super-Yang-Mills

We consider the one-loop finite N = 1 Yang-Mills theories and by a mixture of direct calculation and deductions from the all-orders finiteness of some N = 2 theories we find an expression for the three-loop divergence in the matter field self-energy. We also discuss the effect of explicit supersymmetry breaking terms at the two-loop level.

Thermal self-focusing in a proustite crystal and its thermal properties

The probing beam method was used in an investigation of a steady-state thermal self-focusing in proustite induced by CO2 laser radiation. The dependence of the nonlinear divergence of a probing He–Ne laser beam on the CO2 laser radiation power was determined. The expression for the nonlinear divergence yielded the thermal conductivity. The bulk specific heat was also determined. Estimates were obtained of the critical power and energy needed for the development of self-focusing in proustite.

Divergence of high order Gaussian modes

A simple expression for the beam divergence from a laser oscillating in a high order Gaussian mode can be derived from the properties of the mathematical expressions for these modes.