Infinite Continuum Research Articles

Continuum and Discrete Initial-Boundary Value Problems and Einstein's Field Equations.

Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black-hole problem within Einstein’s theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity.The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial boundary-value problems arising from hyperbolic partial differential equations and to discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein’s equations, and we discuss multi-domain high-order finite difference and spectral methods to solve them.

Chiral and deconfinement transition from Dyson-Schwinger equations

We determine the quark condensate and the dressed Polyakov loop from the finite temperature Landau gauge quark propagator evaluated with U(1)-valued boundary conditions in an approximation to quenched QCD. These gauge invariant quantities allow for an investigation of the chiral and deconfinement transitions. We compare results from Dyson-Schwinger equations on a lattice with infinite volume continuum results and study the temperature and quark mass dependence of both quantities. In particular we investigate the chiral condensate and the dressed Polyakov loop in the chiral limit. We also consider an alternative order parameter for the deconfinement transition, the dual scalar quark dressing, and compare it with the dressed Polyakov loop. As a result we find only slightly different transition temperatures for the chiral and the deconfinement transitions at finite quark masses; in the chiral limit both transitions coincide.

The Re-Wiring of History

The Re-Wiring of History

Three-dimensional physics and the pressure of hot QCD

We update Monte Carlo simulations of the three-dimensional $\mathrm{SU}(3)+\mathrm{\text{adjoint}}$ Higgs theory, by extrapolating carefully to the infinite volume and continuum limits, in order to estimate the contribution of the infrared modes to the pressure of hot QCD. The sum of infrared contributions beyond the known 4-loop order turns out to be a smooth function, of a reasonable magnitude and specific sign. Unfortunately, adding this function to the known 4-loop terms does not improve the match to four-dimensional lattice data, in spite of the fact that other quantities, such as correlation lengths, spatial string tension, or quark number susceptibilities, work well within the same setup. We outline possible ways to reduce the mismatch.

Soil‐structure interaction of tunnel‐structures under consideration of far‐ and near‐field effects

AbstractThe response of a tunnel structure under a moving and oscillating dynamic load can be calculated by means of a hybrid Integral–Transformation–Method–FEM approach. In the ITM approach analytical solutions of ideal geometries of the infinite continuum are superposed such that the wave–number impedance and transfer–function at the interior of a circular tunnel can be calculated efficiently. By this means the infinite continuum can be coupled to an arbitrary tunnel structure which is modelled by the FEM with corresponding form–functions. In order to reduce the effort for the coupling, the characteristics of the analytically given transfer–functions which can be split into near– and far–fields is taken into account. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonlinear interaction between a moving vehicle and a plate elastically mounted on a tunnel

The paper presents a theoretical study of the interaction between a nonlinear model of a moving vehicle (velocity v ) and a plate elastically mounted in a tunnel. An efficient approach to the solution of the problem of vehicle– slab-track– tunnel–soil interaction is developed on the basis of a coupling of the Finite Element and Integral Transform methods (FEM and ITM). According to this approach the tunnel which may have an arbitrary shape and a portion of the surrounding soil is modelled by Finite Elements while the soil (half-space) is described by the ITM. The corresponding solution is found using the solutions for the uniform half-space and for the continuum with a cylindrical cavity. To exploit the invariance of the structure in longitudinal direction x, for this direction additionally to the time–frequency transform ( t ▪ ω ) a space–wavenumber transform ( x ▪ k x ) is used. The case of a half-space is analyzed using a Fourier transform also in the second horizontal direction ( y ▪ k y ) and an analytical solution for the z-direction on the basis of exponential functions. In the case of the infinite continuum with the cavity a Fourier series (for the circumferential direction) and a series of cylindrical functions (for the radial direction) are used for the solution regarding the cross-sectional coordinates ( y , z ) . The tunnel structure, that may have an arbitrary shape, and a portion of the surrounding soil will be modelled by Finite Elements in a ( x ▪ k x , t ▪ ω ) transformed domain. In order to observe the boundary conditions at the surface of the half-space as well as at the surface of the cavity, the superposition of the two solutions has to be performed after an inverse Fourier transform (IFT) in the ( k x , y , z , ω ) -domain. The solution for the complete system floating slab-track– tunnel– half-space is obtained in the ( k x , y , z , ω ) -domain. Once the system transfer function H ( ω ′ ) (with ω ′ = ω + vk x ) for this complete coupled system is found, the displacements of the plate can be calculated in time domain by the IFT of the product p ( ω ′ ) H ( ω ′ ) , i.e. the convolution of the loading p ( t ) with the impulse response function h ( t ) , which completely represents the behavior of the coupled system. In the context of the coupling of systems in a relative movement the problems of differential algebraic equations (DAE) have to be observed.

On the restoration of supersymmetry in twisted two-dimensional lattice Yang–Mills theory

We study a discretization of = 2 super Yang–Mills theory which possesses a single exact supersymmetry at non-zero lattice spacing. This supersymmetry arises after a reformulation of the theory in terms of twisted fields. In this paper we derive the action of the other twisted supersymmetries on the component fields and study, using Monte Carlo simulation, a series of corresponding Ward identities. Our results for SU(2) and SU(3) support a restoration of these additional supersymmetries without fine tuning in the infinite volume continuum limit. Additionally we present evidence supporting a restoration of (twisted) rotational invariance in the same limit. Finally we have examined the distribution of scalar field eigenvalues and find evidence for power law tails extending out to large eigenvalue. We argue that these tails indicate that the classical moduli space does not survive in the quantum theory.

Wave propagation in micropolar mixture of porous media

This paper is concerned with the possible propagation of waves in an infinite porous continuum consisting of a micropolar elastic solid and a micropolar viscous fluid. Micropolar mixture theory of porous media developed by Eringen [A.C. Eringen, Micropolar mixture theory of porous media, J. Appl. Phys. 94 (2003) 4184–4190] is employed. It is found that there exist four coupled longitudinal waves (two coupled longitudinal displacement waves and two coupled longitudinal microrotational waves) and six coupled transverse waves in a continuum of this micropolar mixture. All the waves are found to attenuate and dispersive in nature. A problem of reflection of coupled longitudinal waves from a free boundary surface of a half-space consisting the mixture of a micropolar elastic solid and Newtonian liquid, is investigated. The expressions of various amplitude ratios and surface responses are derived. Numerical computations are performed to find out the phase velocity and attenuation of the waves. The variation of amplitude ratios, energy ratios and surface responses are also computed for a specific model. All the numerical results are depicted graphically. Some limiting cases have also been discussed.

A power flow method for evaluating vibration from underground railways

One of the major sources of ground-borne vibration is the running of trains in underground railway tunnels. Vibration is generated at the wheel–rail interface, from where it propagates through the tunnel and surrounding soil into nearby buildings. An understanding of the dynamic interfaces between track, tunnel and soil is essential before engineering solutions to the vibration problem can be found. A new method has been developed to evaluate the effectiveness of vibration countermeasures. The method is based on calculating the mean power flow from the tunnel, paying attention to that part of the power which radiates upwards to places where buildings’ foundations are expected to be found. The mean power is calculated for an infinite train moving through the tunnel with a constant velocity. An elegant mathematical expression for the mean power flow is derived, which can be used with any underground-tunnel model. To evaluate the effect of vibration countermeasures and track properties on power flow, a comprehensive three-dimensional analytical model is used. It consists of Euler–Bernoulli beams to account for the rails and the track slab. These are coupled in the wavenumber–frequency domain to a thin shell representing the tunnel embedded within an infinite continuum, with a cylindrical cavity representing the surrounding soil.

Wave propagation and localisation in a softening two-phase medium

Within the framework of the generalised theory of heterogeneous media, the complete set of equations is derived for a three-dimensional fluid-saturated porous medium. Subsequently, stability and dispersion analyses are carried out for an infinite one-dimensional continuum that has been deforming homogeneously prior to the application of the perturbation. For common values of the stiffness, the porosity and the mass densities, the two-phase medium is found to remain stable when the solid is elastic. When an elastoplastic or an elasticity-based damage model is adopted for the solid constituent, loss of stability occurs at the peak in the uniaxial stress–strain relation for the solid constituent, which is identical to a single-phase medium. From the dispersion analysis it appears that a dispersive wave is obtained, but that the internal length scale associated with it vanishes in the short wavelength limit, at least for the assumptions made regarding the constitutive behaviour of the solid and of the fluid. This result leads to the conclusion that, upon the introduction of softening, localisation in a zero width will occur and no regularisation will be present. This conclusion is corroborated by the results of numerical analyses of wave propagation in a finite one-dimensional bar.

Interaction of a One-Dimensional Continuum with an Inertial Object Moving over the Continuum

Free transverse oscillations in a system consisting of an infinite moment continuum, such as the Euler-Bernoulli beam lying on the Winkler foundation, and a rigid body moving along the beam with a constant velocity and having a point contact with the guide are studied. The range of the considered velocities of the concentrated inertial object along the continuum is limited by the requirement of a finite energy of elastic deformation of the infinite continuum, corresponding to cojoint free osillations of an unbounded system. An analytical solution of the corresponding spectral problem in a system with a mixed spectrum is constructed. Limiting situations are analyzed, where the inertial rigid object moving along the beam is devoid of one “oscillatory” degree of freedom for some reasons. In particular, an inertial object devoid of mass but having a nonzero tensor of inertia is considered. Dependences of all characteristics of the discrete spectrum of oscillations and their shapes on the magnitude of object velocity along the moment elastoinertial guide are given.

LATTICE STUDY ON KAON–NUCLEON SCATTERING LENGTH IN THE I=1 CHANNEL

Using the tadpole improved clover Wilson quark action on small, coarse and anisotropic lattices, KN scattering length in the I=1 channel is calculated within quenched approximation. The results are extrapolated towards the chiral and physical kaon mass region. Finite volume and finite lattice spacing errors are also analyzed and a result in the infinite volume and continuum limit is obtained which is compatible with the experiment and the results from Chiral Perturbation Theory.

Lattice study on kaon–pion scattering length in the I=3/2 channel

Using the tadpole improved Wilson quark action on small, coarse and anisotropic lattices, Kπ scattering length in the I=3/2 channel is calculated within quenched approximation. The results are extrapolated towards the chiral and physical kaon mass region. Finite volume and finite lattice spacing errors are also analyzed and a result in the infinite volume and continuum limit is obtained. Our result is compared with the results obtained using Roy equations, chiral perturbation theory, dispersion relations and the experimental data.

THE PHYSICAL BASIS FOR PARRONDO'S GAMES

It has been reported that the original inspiration for Parrondo's games was a physical system called a "flashing Brownian ratchet." The relationship seems to be intuitively plausible but has not previously established with rigor. This is the problem that we address in this paper. The dynamics of a Brownian particle in a flashing Brownian ratchet are the result of diffusion and of externally applied forces. The probability density, of finding the particle at a certain place and time, can be mathematically modelled using a Partial Differential Equation (PDE) namely the Fokker-Planck Equation. In this paper, we apply standard finite-difference methods of numerical analysis to the Fokker-Planck Equation. We derive a set of finite difference equations and show that they have the same form as Parrondo's games which justifies the claim that Parrondo's games are a discrete-time, discrete-space version of a flashing Brownian ratchet. We claim that Parrondo's games are effectively a particular way of sampling a Fokker-Planck Equation. Our difference equations are a natural and physically motivated generalization of Parrondo's games. We refer to some well established theorems of numerical analysis to suggest conditions under which the solutions to the difference equations and partial differential equations would converge to the same solution. The diffusion operator, implicitly assumed in Parrondo's original games, reduces to a pre-existing numerical method called "The Schmidt formula." There is actually an infinite continuum of possible diffusion operators and the Schmidt formula is at one extreme of the feasible range. We point out that an operator in the middle of the feasible range, with half-period binomial weightings, would be a better representation of the underlying physics. We present a modified form of Parrondo's games based on the central diffusion operator. We suggest that the finite difference method presented here will be useful in the simulation and design of real physical flashing Brownian ratchets.

Order of the metal-to-superconductor transition

We present results from large-scale Monte Carlo simulations on the full Ginzburg-Landau (GL) model, including fluctuations in the amplitude and the phase of the matter field, as well as fluctuations of the noncompact gauge field of the theory. From this we obtain a precise critical value of the GL parameter ${\ensuremath{\kappa}}_{\mathrm{tri}}$ separating a first-order metal-to-superconductor transition from a second-order one, ${\ensuremath{\kappa}}_{\mathrm{tri}}=(0.76\ifmmode\pm\else\textpm\fi{}0.04)/\sqrt{2}.$ This agrees surprisingly well with earlier analytical results based on a disorder theory of the superconductor to metal transition, where the value ${\ensuremath{\kappa}}_{\mathrm{tri}}=0.798/\sqrt{2}$ was obtained. To achieve this, we have done careful infinite volume and continuum limit extrapolations. In addition we offer an interpretation of ${\ensuremath{\kappa}}_{\mathrm{tri}},$ namely, that it is also the value separating type-I and type-II behavior.

Calculating the I=2 pion scattering length using tadpole improved clover Wilson action on coarse anisotropic lattices

In an exploratory study, using the tadpole improved clover Wilson quark action on small, coarse and anisotropic lattices, the ππ scattering length in the I=2 channel is calculated within quenched approximation. A new method is proposed which enables us to make chiral extrapolation of our lattice results without calculating the decay constant on the lattice. Finite volume and finite lattice spacing errors are analyzed and the results are extrapolated towards the infinite volume and continuum limit. Comparisons of our lattice results with the new experiment and the results from Chiral Perturbation Theory are made. Good agreements are found.

Infinite volume and continuum limits of the Landau-gauge gluon propagator

We extend a previous improved action study of the Landau gauge gluon propagator, by using a variety of lattices with spacings from $a = 0.17$ to 0.41 fm, to more fully explore finite volume and discretization effects. We also extend a previously used technique for minimizing lattice artifacts, the appropriate choice of momentum variable or ``kinematic correction'', by considering it more generally as a ``tree-level correction''. We demonstrate that by using tree-level correction, determined by the tree-level behavior of the action being considered, it is possible to obtain scaling behavior over a very wide range of momenta and lattice spacings. This makes it possible to explore the infinite volume and continuum limits of the Landau-gauge gluon propagator.

A molecular dynamics investigation on grain disappearance at a triple junction in polycrystalline silicon

ABSTRACTTopological changes in microstructure are strictly related to the microscopic evolution of triple junctions (TJ). The three-sided grain disappearance, usually called T2 process, is here investigated via 3D-atomistic modeling. In particular the stability of a three-sided grain insertion in a triple junction in silicon is studied within the framework of Molecular Dynamics simulations. The Stillinger-Weber interatomic potential is adopted and constant-traction border conditions are considered to ensure a proper embedding of the atomistic region in a virtually infinite bulk continuum. Dealing with the T2-event, the critical radius below which the three- sided inner grain become unstable is evaluated to be three to four times the lattice constant of silicon. Moreover, we show that the instability sets in through the amorphization of the central shrinking grain.

Small perturbation method of high-frequency bistatic volume scattering from marine sediments

The first-order small perturbation method (Born approximation), as frequently applied to high-frequency scattering from marine sediments, is critically reviewed, tested for accuracy and extended. The sediment is modeled as an acoustic-fluid half space with random fluctuations in density and compressibility. Several cases of volume scattering from typical marine sand and mud sediments are presented to illustrate the effects of two important assumptions: (1) the effects of assuming the density and compressibility fluctuations are proportional, and (2) half-space effects. By relaxing the assumption that the sediment density and compressibility are proportional, the bistatic scattering cross section is significantly altered. The effects of properly modeling the sediment as a random half space (as opposed to an infinite continuum) are also discussed. In the context of first-order perturbation theory, half-space effects manifest themselves as a "modified" spectra for density and compressibility fluctuations. It is shown that, for lossy sediments and for scattering near the specular direction, half-space effects are significant and cannot be neglected. This result is significant because current models of sediment volume scattering do not include half-space effects. In addition to the theoretical model, exact numerical simulations are used to evaluate the accuracy of the perturbation model for a limited number of cases.

Electroweak phase transition in the minimal supersymmetric standard model: 4-dimensional lattice simulations

Lattice results show no standard model (SM) electroweak phase transition (EWPT) for Higgs masses approximately 72 GeV, which is below the present experimental limit. Perturbation theory and 3-dimensional simulations indicate an EWPT in the minimal supersymmetric SM (MSSM) that is strong enough for baryogenesis up to m(h) approximately 105 GeV. In this Letter we present the results of our large scale 4-dimensional MSSM EWPT simulations. We carried out infinite volume and continuum limits and found a transition whose strength agrees well with perturbation theory, allowing MSSM electroweak baryogenesis at least up to m(h) = 103+/-4 GeV. We determined the properties of the bubble wall.