Poincaré gauge theory with coupled even and odd parity spin‐0 modes: Cosmological normal modes

We are investigating the dynamics of a new Poincaré gauge theory of gravity model, which has cross coupling between the spin‐0⁺ and spin‐0^‐ modes. To this end we are considering a very appropriate situation – homogeneous‐isotropic cosmologies – which is relatively simple, and yet all the modes have non‐trivial dynamics which reveals physically interesting and possibly observable results. More specifically we consider manifestly isotropic Bianchi class A cosmologies. Here the first order equations obtained from an effective Lagrangian are linearized and the normal modes are found. These turn out to control the asymptotic late time cosmological normal modes. Numerical evolution confirms the late time asymptotic approximation and shows the expected effects of the cross parity pseudoscalar coupling.     

Asymptotic Regimes of Magnetic Bianchi Cosmologies

We consider the asymptotic dynamics of the Einstein-Maxwell field equations for the class of non-tilted Bianchi cosmologies with a barotropic perfect fluid and a pure homogeneous source-free magnetic field, with emphasis on models of Bianchi type VII₀, which have not been previously studied. Using the orthonormal frame formalism and Hubble-normalized variables, we show that, as is the case for the previously studied class A magnetic Bianchi models, the magnetic Bianchi VII₀ cosmologies also exhibit an oscillatory approach to the initial singularity. However, in contrast to the other magnetic Bianchi models, we rigorously establish that typical magnetic Bianchi VII₀ cosmologies exhibit the phenomena of asymptotic self-similarity breaking and Weyl curvature dominance in the late-time regime.     

Cosmologies with variable Newton's “constant”

We discuss cosmologies where the Planck length is not a fundamental constant but rather evolves with time. The dynamics which should be responsible for today's tiny value of this length scale are governed by the effective potential of a Brans-Dicke type theory. Qualitative properties of this potential depend on the short distance behaviour of the unifying fundamental theory. We discuss criteria for the asymptotic behaviour of realistic cosmologies and show that the role of a possible cosmological constant is quite different from the case of standard cosmology.     

Asymptotic Behavior for Rayleigh Problem Based on Kinetic Theory

We investigate the dynamics of the gas bounded by an infinite flat plate which is initially in equilibrium and set at some instant impulsively into uniform motion in its own plane. We use the Boltzmann equation to describe intermolecular collisions and assume the diffuse reflection to describe the interaction of the gas with the boundary. The Mach number of the plate is assumed to be small so that we can linearize the Boltzmann equation as well as the boundary condition. We show that the asymptotic behavior of the gas represents a perturbation to the free molecular gas when the time is much less than the mean free time. On the other hand, if the time is much greater than the mean free time, we show that the gas dynamics is governed by the linearized Navier–Stokes equation with a slip flow on the boundary and establish a boundary layer correction with thickness of the order of the mean free path. We also establish the singularity of velocity distribution function along the particle trajectory near the boundary.     

Thermodynamics of Universe with Time Varying Cosmological Term

The FRW-type cosmologies with time varying cosmological term is discussed within the frame work of a thermodynamic context. If at some cosmological time, the cosmological term begins increasing again, as presently observed, expansion will accelerate and matter and/or radiation will be transformed back into dark energy. It is shown that such accelerated expansion is a route towards a new kind of gravitational singular state, characterized by an empty, conformally transitive spacetime in which all energy is dark. We investigate whether dynamic dark energy cosmologies are compatible with the second law of thermodynamics. We examine also the total entropy evolution with time. We observed that the dynamic dark energy cosmology is less restricted with second law of thermodynamics. Some physical implications of these solutions are briefly discussed.     

On multimeron solutions of the Yang-Mills equation

We study a singular boundary value problem introduced by Glimm and Jaffe for the purpose of obtaining solutions of the Euclidean Yang-Mills equations with isolated singularities along an axis. Using comparison techniques, we prove existence, asymptotic behavior and also uniqueness in some special cases.     

Accurate computation of Stokes flow driven by an open immersed interface

We present numerical methods for computing two-dimensional Stokes flow driven by forces singularly supported along an open, immersed interface. Two second-order accurate methods are developed: one for accurately evaluating boundary integral solutions at a point, and another for computing Stokes solution values on a rectangular mesh. We first describe a method for computing singular or nearly singular integrals, such as a double layer potential due to sources on a curve in the plane, evaluated at a point on or near the curve. To improve accuracy of the numerical quadrature, we add corrections for the errors arising from discretization, which are found by asymptotic analysis. When used to solve the Stokes equations with sources on an open, immersed interface, the method generates second-order approximations, for both the pressure and the velocity, and preserves the jumps in the solutions and their derivatives across the boundary. We then combine the method with a mesh-based solver to yield a hybrid method for computing Stokes solutions at N² grid points on a rectangular grid. Numerical results are presented which exhibit second-order accuracy. To demonstrate the applicability of the method, we use the method to simulate fluid dynamics induced by the beating motion of a cilium. The method preserves the sharp jumps in the Stokes solution and their derivatives across the immersed boundary. Model results illustrate the distinct hydrodynamic effects generated by the effective stroke and by the recovery stroke of the ciliary beat cycle.     

Existence and Dynamics of Bounded Traveling Wave Solutions to Getmanou Equation

In this paper, we study the existence and dynamics of bounded traveling wave solutions to Getmanou equations by using the qualitative theory of differential equations and the bifurcation method of dynamical systems. We show that the corresponding traveling wave system is a singular planar dynamical system with two singular straight lines, and obtain the bifurcations of phase portraits of the system under different parameters conditions. Through phase portraits, we show the existence and dynamics of several types of bounded traveling wave solutions including solitary wave solutions, periodic wave solutions, compactons, kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions are given. Additionally, we confirm abundant dynamical behaviors of the traveling wave s olutions to the equation, which are summarized as follows: i) We confirm that two types of orbits give rise to solitary wave solutions, that is, the homoclinic orbit passing the singular point, and the composed homoclinic orbit which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of associated system. ii) We confirm that two types of orbits correspond to periodic wave solutions, that is, the periodic orbit surrounding a center, and the homoclinic orbit of associated system, which is tangent to the singular line at the singular point of associated system.     

Asymptotics of trajectories for cone potentials

We study the asymptotics of trajectories of the classical Hamiltonian dynamics. For Hamiltonians with cone potentials we have shown earlier that all trajectories are asymptotically free [5], i.e. the asymptotic velocities exist. Here we show that the generic trajectories are asymptotically uniform, i.e. the asymptotic phases exist.     

Dynamics of locally rotationally symmetric Bianchi type VIII cosmologies with anisotropic matter

This paper is a study of the effects of anisotropic matter sources on the qualitative evolution of spatially homogenous cosmologies of Bianchi type VIII. The analysis is based on a dynamical system approach and makes use of an anisotropic matter family developed by Calogero and Heinzle which generalises perfect fluids and provides a measure of deviation from isotropy. Thereby the role of perfect fluid solutions is put into a broader context. The results of this paper concern the past and future asymptotic dynamics of locally rotationally symmetric solutions of type VIII with anisotropic matter. It is shown that solutions whose matter source is sufficiently close to being isotropic exhibit the same qualitative dynamics as perfect fluid solutions. However a high degree of anisotropy of the matter model can cause dynamics to differ significantly from the vacuum and perfect fluid case.     

Asymptotic analysis of Bloch-Floquet waves in a thin bi-material strip with a periodic array of finite-length cracks

We present an asymptotic algorithm to solve a problem of wave propagation in a thin bi-material strip with an array of cracks situated at the interface between two materials. For small frequencies we construct an asymptotic solution which takes into account the singular behavior near the crack tips and the smooth nature of the oscillation far away from them. We construct the boundary layer solutions near the crack tips. The boundary layers are harmonic solutions in scaled domains. Dispersion equations are derived and solved within the frame of the asymptotic model.     

Asymptotic analysis of Bloch–Floquet waves in a thin bi-material strip with a periodic array of finite-length cracks

We present an asymptotic algorithm to solve a problem of wave propagation in a thin bi-material strip with an array of cracks situated at the interface between two materials. For small frequencies we construct an asymptotic solution which takes into account the singular behavior near the crack tips and the smooth nature of the oscillation far away from them. We construct the boundary layer solutions near the crack tips. The boundary layers are harmonic solutions in scaled domains. Dispersion equations are derived and solved within the frame of the asymptotic model.     

Renormalization Group Approach to Generalized Cosmological Models

We revisit here the problem of generalized cosmology using renormalization group approach. A complete analysis of these cosmologies, where specific models appear as asymptotic fixed-points, is given here along with their linearized stability analysis.     

On the elliptic mesh generation in domains containing multiple inclusions and undergoing large deformations

We present an improved method to generate a sequence of structured meshes even when the physical domain contains deforming inclusions. This method belongs to the class of Arbitrary Lagrangian–Eulerian (ALE) methods for solving moving boundary problems. Its tools are either (a) separate mappings of the domain boundaries and enforcing the node distribution on lines emanating from singular points or (b) domain decomposition and separate mappings of each subdomain using suitable coordinate systems. The latter is shown to be more versatile and general. In both cases a set of elliptic equations is used to generate the grid extending in this way the method advanced by Dimakopoulos and Tsamopoulos [Y. Dimakopoulos, J.A. Tsamopoulos, A quasi-elliptic transformation for moving boundary problems with large anisotropic deformations, J. Comput. Phys. 192 (2003) 494–522]. We shall present examples where this earlier method and all other mesh generating methods which are based on a conformal mapping or solving a quasi-elliptic set of PDEs fail to produce an acceptable mesh and accurate solutions in such geometries. Furthermore, in contrast to other methods, appropriate boundary conditions and constraints such as, orthogonality of specific mesh lines and prespecified node distributions on them, can be easily implemented along a specific part of the domain or its boundary. Hence, no attractive terms at specific corners or singular points are needed. To increase the mesh resolution around the moving interfaces while keeping low the memory requirements and the computational time, a local mesh refinement technique has been incorporated as well. The method is demonstrated in two challenging examples where no remeshing is required in spite of the large domain deformations. In the first one, the transient growth of two bubbles embedded in a viscoelastic filament undergoing stretching in the axial direction is examined, while in the second one the linear and non-linear dynamics of two bubbles in a viscous medium are determined in an acoustic field. The large elasticity of the filament in the first case or the large inertia in the second case coupled with the externally induced large deformations of the liquid domain requires the accurate calculation which is achieved by the method we propose herein. The governing equations are solved using the finite element/Galerkin method with appropriate modifications to solve the hyperbolic constitutive equation of a viscoelastic fluid. These are coupled with an implicit Euler method for time integration or with Arnoldi’s algorithm for normal mode analysis.     

Unruh–DeWitt Detectors in Spherically Symmetric Dynamical Space-Times

In the present paper, Unruh–DeWitt detectors are used in order to investigate the issue of temperature associated with a spherically symmetric dynamical space-times. Firstly, we review the semi-classical tunneling method, then we introduce the Unruh–DeWitt detector approach. We show that for the generic static black hole case and the FRW de Sitter case, making use of peculiar Kodama trajectories, semiclassical and quantum field theoretic techniques give the same standard and well known thermal interpretation, with an associated temperature, corrected by appropriate Tolman factors. For a FRW space-time interpolating de Sitter space with the Einstein–de Sitter universe (that is a more realistic situation in the frame of ΛCDM cosmologies), we show that the detector response splits into a de Sitter contribution plus a fluctuating term containing no trace of Boltzmann-like factors, but rather describing the way thermal equilibrium is reached in the late time limit. As a consequence, and unlike the case of black holes, the identification of the dynamical surface gravity of a cosmological trapping horizon as an effective temperature parameter seems lost, at least for our co-moving simplified detectors. The possibility remains that a detector performing a proper motion along a Kodama trajectory may register something more, in which case the horizon surface gravity would be associated more likely to vacuum correlations than to particle creation.     

Asymptotic evolution of random unitary operations

We analyze the asymptotic dynamics of quantum systems resulting from large numbers of iterations of random unitary operations. Although, in general, these quantum operations cannot be diagonalized it is shown that their resulting asymptotic dynamics is described by a diagonalizable superoperator. We prove that this asymptotic dynamics takes place in a typically low dimensional attractor space which is independent of the probability distribution of the unitary operations applied. This vector space is spanned by all eigenvectors of the unitary operations involved which are associated with eigenvalues of unit modulus. Implications for possible asymptotic dynamics of iterated random unitary operations are presented and exemplified in an example involving random controlled-not operations acting on two qubits.     

Static,Spherically Symmetric Solutions of Yang-Mills-Dilaton Theory

Static, spherically symmetric solutions of the Yang-Mills-Dilaton theory are studied. It is shown that these solutions fall into three different classes. The generic solutions are singular. Besides there is a discrete set of globally regular solutions further distinguished by the number of nodes of their Yang-Mills potential. The third class consists of oscillating solutions playing the role of limits of regular solutions, when the number of nodes tends to infinity. We show that all three sets of solutions are non-empty. Furthermore we give asymptotic formulae for the parameters of regular solutions and confront them with numerical results.     

Asymptotic behaviors of the stress fields in the vicinity of dislocations and dislocation segments

We obtain the singular asymptotic behavior of the stress field in the vicinity of a non-planar dislocation in three dimensions and the nearly singular behavior of the full self-force of the dislocation including both glide and climb forces, using asymptotic analysis. We also derive asymptotic formulas for the stress field in the vicinity of a curved dislocation segment. Numerical examples are presented to examine the asymptotic formulas. The obtained formulas can be used for qualitative understanding of the stress tensor associated with dislocations and efficient and accurate calculation of the stress tensor in dislocation dynamics simulations.     

Supersymmetry of FRW Barotropic Cosmologies

Barotropic FRW cosmologies are presented from the standpoint of nonrelativistic supersymmetry. First, we reduce the barotropic FRW system of differential equations to simple harmonic oscillator differential equations. Employing the factorization procedure, the solutions of the latter equations are divided into the two classes of bosonic (nonsingular) and fermionic (singular) cosmological solutions. We next introduce a coupling parameter denoted by K between the two classes of solutions and obtain barotropic cosmologies with dissipative features acting on the scale factors and spatial curvature of the universe. The K-extended FRW equations in comoving time are presented in explicit form in the low coupling regime. The standard barotropic FRW cosmologies correspond to the dissipationless limit K = 0.