Phase Diagram of a Two-Species Lattice Model with a Linear Instability

We discuss the properties of a one-dimensional lattice model of a driven system with two species of particles in which the mobility of one species depends on the density of the other. This model was introduced by Lahiri and Ramaswamy ( Phys . Rev. Lett ., 79 , 1150 (1997)) in the context of sedimenting colloidal crystals, and its continuum version was shown to exhibit an instability arising from linear gradient couplings. In this paper we review recent progress in understanding the full phase diagram of the model. There are three phases. In the first, the steady state can be determined exactly along a representative locus using the condition of detailed balance. The system shows phase separation of an exceptionally robust sort, termed strong phase separation, which survives at all temperatures. The second phase arises in the threshold case where the first species evolves independently of the second, but the fluctuations of the first influence the evolution of the second, as in the passive scalar problem. The second species then shows phase separation of a delicate sort, in which long-range order coexists with fluctuations which do not damp down in the large-size limit. This fluctuation-dominated phase ordering is associated with power law decays in cluster size distributions and a breakdown of the Porod law. The third phase is one with a uniform overall density, and along a representative locus the steady state is shown to have product measure form. Density fluctuations are transported by two kinematic waves, each involving both species and coupled at the nonlinear level. Their dissipation properties are governed by the symmetries of these couplings, which depend on the overall densities. In the most interesting case, the dissipation of the two modes is characterized by different critical exponents, despite the nonlinear coupling.     

Three-dimensional finite-difference lattice Boltzmann model and its application to inviscid compressible flows with shock waves

In this paper, a three-dimensional (3D) finite-difference lattice Boltzmann model for simulating compressible flows with shock waves is developed in the framework of the double-distribution-function approach. In the model, a density distribution function is adopted to model the flow field, while a total energy distribution function is adopted to model the temperature field. The discrete equilibrium density and total energy distribution functions are derived from the Hermite expansions of the continuous equilibrium distribution functions. The discrete velocity set is obtained by choosing the abscissae of a suitable Gauss–Hermite quadrature with sufficient accuracy. In order to capture the shock waves in compressible flows and improve the numerical accuracy and stability, an implicit–explicit finite-difference numerical technique based on the total variation diminishing flux limitation is introduced to solve the discrete kinetic equations. The model is tested by numerical simulations of some typical compressible flows with shock waves ranging from 1D to 3D. The numerical results are found to be in good agreement with the analytical solutions and/or other numerical results reported in the literature.     

Lattice Boltzmann method on quadtree grids for simulating fluid flow through porous media: A new automatic algorithm

During the past two decades, the lattice Boltzmann (LB) method has been introduced as a class of computational fluid dynamic methods for fluid flow simulations. In this method, instead of solving the Navier Stocks equation, the Boltzmann equation is solved to simulate the flow of a fluid. This method was originally developed based on uniform grids. However, in order to model complex geometries such as porous media, it can be very slow in comparison with other techniques such as finite differences and finite elements. To eliminate this limitation, a number of studies have aimed to formulate the lattice Boltzmann on the unstructured grids. This paper deals with simulating fluid flow through a synthetic porous medium using the LB method and on the quadtree grid structure. To this end, the LB method was used on nonuniform grids coupled with a technique for image reconstruction which resulted in the quadtree grids for simulation of fluid flow through porous media. Accuracy and efficiency of this algorithm is compared against the conventional LB method based on uniform grids. While the decrease in computational time in the proposed LB method on nonuniform grids is found to be significant regarding the size of the initial and reconstructed images, the same level of accuracy is obtained when compared with the conventional LB method on uniform grids.     

Direct evidence for a Coulombic phase in monopole-suppressed SU(2) lattice gauge theory

Further evidence is presented for the existence of a non-confining phase at weak coupling in SU(2) lattice gauge theory. Using Monte Carlo simulations with the standard Wilson action, gauge-invariant SO(3)–Z2 monopoles, which are strong-coupling lattice artifacts, have been seen to undergo a percolation transition exactly at the phase transition previously seen using Coulomb gauge methods, with an infinite lattice critical point near β=3.2$\beta =3.2$. The theory with both Z2 vortices and monopoles and SO(3)–Z2 monopoles eliminated is simulated in the strong-coupling (β=0$\beta =0$) limit on lattices up to 60⁴. Here, as in the high-β phase of the Wilson-action theory, finite size scaling shows it spontaneously breaks the remnant symmetry left over after Coulomb gauge fixing. Such a symmetry breaking precludes the potential from having a linear term. The monopole restriction appears to prevent the transition to a confining phase at any β  . Direct measurement of the instantaneous Coulomb potential shows a Coulombic form with moderately running coupling possibly approaching an infrared fixed point of α∼1.4$\alpha \sim 1.4$. The Coulomb potential is measured to 50 lattice spacings and 2 fm. A short-distance fit to the 2-loop perturbative potential is used to set the scale. High precision at such long distances is made possible through the use of open boundary conditions, which was previously found to cut random and systematic errors of the Coulomb gauge fixing procedure dramatically. The Coulomb potential agrees with the gauge-invariant interquark potential measured with smeared Wilson loops on periodic lattices as far as the latter can be practically measured with similar statistics data.     

A solution to the non-Abelian duality problem

Dualities uniquely excel at resolving non-perturbative aspects of complex phase diagrams of interacting, Landau or topologically ordered, systems. However, traditional duality transformations fail for systems like the Heisenberg model and non-Abelian gauge theories. The bond-algebraic theory of quantum and classical dualities provides a solution to this long-standing conundrum, the so-called non-Abelian duality problem, by embedding traditional dualities into a more general transformation scheme that always preserves locality in any number of dimensions. Remarkably, it turns out to be unimportant whether a model?s group of symmetries is Abelian or non-Abelian. The capability of the bond-algebraic approach to handle finite and infinite systems with arbitrary boundary conditions has recently led to the discovery of holographic symmetries  , relating topological order, edge states, and generalized order parameters. We discuss the interplay between these distinguished boundary symmetries and our solution to the non-Abelian duality problem. To illustrate our technique we present, among others, novel dualities for the SU(2)$\mathit{SU}(2)$ principal chiral field and both U(1)$U(1)$ and SU(2)$\mathit{SU}(2)$ generalizations of the planar quantum compass model of orbital ordering.     

MgB2 and AlB2 at high pressures

We have studied the effect of pressure on vibrational and electronic properties of MgB₂ and AlB₂ by ab initio calculations and Raman spectroscopy. The comparison between the calculations and the Raman data puts the common assignment of a broad spectral feature near 600 cm^?1 in MgB₂ to the E _2g phonon into question. At variance with MgB₂ the Raman spectra of AlB₂ exhibit a well-defined E _2g mode indicating that the anomalous Raman spectrum encountered in MgB₂ is not related to the metallicity of the samples nor is it intrinsic to crystals of the AlB₂ structure type. A theoretical estimate of the pressure dependence of T _c in MgB₂ shows that the experimentally observed decrease of T _c under pressure is predominantly due to phonon frequency shifts.     

Inelastic X-Ray scattering in cs under pressure

The pressure dependences of the longitudinal acoustic (LA), Q