The onset of convection in a couple stress fluid saturated porous layer using a thermal non-equilibrium model

The stability of a couple stress fluid saturated horizontal porous layer heated from below and cooled from above when the fluid and solid phases are not in local thermal equilibrium is investigated. The Darcy model is used for the momentum equation and a two-field model is used for energy equation each representing the solid and fluid phases separately. The linear stability theory is employed to obtain the condition for the onset of convection. The effect of thermal non-equilibrium on the onset of convection is discussed. It is shown that the results of the thermal non-equilibrium Darcy model for the Newtonian fluid case can be recovered in the limit as couple stress parameter C→0. We also present asymptotic analysis for both small and large values of the inter phase heat transfer coefficient H. We found an excellent agreement between the exact solutions and asymptotic solutions when H is very small.     

R-matrix approach to the 3-body problem

The asymptotic form of the Faddeev amplitude in coordinate space is derived in various orders. This form and the structure of the Faddeev equations allow by aR-matrix method to establish a set of equations directly for the 3-body on-shellT-matrix elements. The procedure is equally well suited for local and nonlocal interactions.     

Evaluation of functional integrals for an ultralocal static field theory

Exact and perturbative solutions of an ultralocal, static, real φ^m field theory, obtained by functional integral methods are given and discussed. It is shown that this unrenormalizable theory leads to a free one in the local limit, whereas a small non-locality leads to a small effective coupling constant in an effective φ⁴ Lagrangian. It is proved that the usual perturbation seeries is the asymptotic expansion of the exact result.     

Multichannel framework for singular quantum mechanics

A multichannel S-matrix framework for singular quantum mechanics (SQM) subsumes the renormalization and self-adjoint extension methods and resolves its boundary-condition ambiguities. In addition to the standard channel accessible to a distant (“asymptotic”) observer, one supplementary channel opens up at each coordinate singularity, where local outgoing and ingoing singularity waves coexist. The channels are linked by a fully unitary S-matrix, which governs all possible scenarios, including cases with an apparent nonunitary behavior as viewed from asymptotic distances.     

Radial asymptotics of Lemaître–Tolman–Bondi dust models

We examine the radial asymptotic behavior of spherically symmetric Lemaître–Tolman–Bondi dust models by looking at their covariant scalars along radial rays, which are spacelike geodesics parametrized by proper length ?, orthogonal to the 4-velocity and to the orbits of SO(3). By introducing quasi-local scalars defined as integral functions along the rays, we obtain a complete and covariant representation of the models, leading to an initial value parametrization in which all scalars can be given by scaling laws depending on two metric scale factors and two basic initial value functions. Considering regular “open” LTB models whose space slices allow for a diverging ?, we provide the conditions on the radial coordinate so that its asymptotic limit corresponds to the limit as ? → ∞. The “asymptotic state” is then defined as this limit, together with asymptotic series expansion around it, evaluated for all metric functions, covariant scalars (local and quasi-local) and their fluctuations. By looking at different sets of initial conditions, we examine and classify the asymptotic states of parabolic, hyperbolic and open elliptic models admitting a symmetry center. We show that in the radial direction the models can be asymptotic to any one of the following spacetimes: FLRW dust cosmologies with zero or negative spatial curvature, sections of Minkowski flat space (including Milne’s space), sections of the Schwarzschild–Kruskal manifold or self-similar dust solutions.     

Evaluation of the spin wave Green's functions for rutile structure Heisenberg antiferromagnets with exchange between ions both on the same and opposite sublattices

A straightforward method is presented for the evaluation of the spin wave Green's function appropriate to the Raman scattering in the xy and xz geometry in rutile structure Heisenberg antiferromagnets with exchange between ions both on the same and on opposite sublattices and arbitrary local anisotropy. The analytical asymptotic behaviour of the Green's functions near the singularities is explicitly given and the problem of the numerical evaluation of their real and imaginary parts is discussed. Tables of the imaginary parts, calculated at the points corresponding to a Gaussian quadrature procedure in the appropriate interval, are supplied on request.     

On the thermodynamics of the random one-dimensional Ising chain in a transverse field

For three simple one-dimensional disordered models: (a) the Ising chain with random magnetic moments in a transverse field, (b) the Ising chain with random coupling constants in a transverse field, and (c) the X-Y model with a special type of disorder, the asymptotic equivalence in the thermodynamic limit is proved and some of its consequences are discussed. The spectral density of the finite chain for the model (a) is calculated by Dean's method for several representative cases and the presence of the local modes is indicated. The expressions for the initial susceptibilities for the models (a) and (b) are reviewed and (in two cases) the derivations are simplified.     

Higher-order correlation functions of the planar ising model

A method of calculating the asymptotic behaviour of the higher-order correlation functions for large distances is proposed for the planar Ising model in the absence of a magnetic field. The three-point correlation functions composed of a spin operator or of energy-density operators are considered. The asymptotic behaviour of the correlation functions for distances R ? R_c (where R_c is the correlation radius) is determined. It is shown that the asymptotic behaviour of the correlation functions for large distances does not depend on the choice of operators. The asymptotic behaviour of the correlation functions in which two operators are relatively close to one another is considered near the critical point. The results which we obtained are compared with the predictions of the scaling laws and operator algebra.     

Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations

We study the asymptotic behavior and the asymptotic stability of the 2D Euler equations and of the 2D linearized Euler equations close to parallel flows. We focus on flows with spectrally stable profiles U(y) and with stationary streamlines y=y₀ (such that U^′(y₀)=0), a case that has not been studied previously. We describe a new dynamical phenomenon: the depletion of the vorticity at the stationary streamlines. An unexpected consequence is that the velocity decays for large times with power laws, similarly to what happens in the case of the Orr mechanism for base flows without stationary streamlines. The asymptotic behaviors of velocity and the asymptotic profiles of vorticity are theoretically predicted and compared with direct numerical simulations. We argue on the asymptotic stability of this ensemble of flow profiles even in the absence of any dissipative mechanisms.     

Ensemble Dependence of Fluctuations: Canonical Microcanonical Equivalence of Ensembles

We study the equivalence of microcanonical and canonical ensembles in continuous systems, in the sense of the convergence of the corresponding Gibbs measures and the first order corrections. We are particularly interested in extensive observables, like the total kinetic energy. This result is obtained by proving an Edgeworth expansion for the local central limit theorem for the energy in the canonical measure, and a corresponding local large deviations expansion. As an application we prove a formula due to Lebowitz–Percus–Verlet that express the asymptotic microcanonical variance of the kinetic energy in terms of the heat capacity.     

Charged sectors and scattering states in quantum electrodynamics

Charge operator and charged sectors in four-dimensional QED are investigated, both within a local, covarian (indefinite metric), and within a positive metric formulation of QED. Assuming a few basic, physical principles-such as Gauss'law and a correspondence principle, etc.-we conclude that charge states determine non-Fock (coherent state) representations of the algebra generated by the asymptotic, electromagnetic field, that Lorentz boosts do not leave the harged sectors invariant (spontaneous breaking of boost symmetry), and that an unambiguous definition of the “mass” of the charged infraparticles is possible. This and other results represent first steps at extending the Haag-Ruelle scattering theory to charged infraparticles.     

Collision theory for waves in two dimensions and a characterization of models with trivialS-matrix

Within the framework of local relativistic quantum theory in two space-time dimensions, we develop a collision theory for waves (the set of vectors corresponding to the eigenvalue zero of the mass operator). Since among these vectors there need not be one-particle states, the asymptotic Hilbert spaces do not in general have Fock structure. However, the definition and “physical interpretation” of anS-matrix is still possible. We show that thisS-matrix is trivial if the correlations between localized operators vanish at large timelike distances.     

Generalized asymptotic expansions for coupled wavenumbers in fluid-filled cylindrical shells

Analytical expressions are found for the coupled wavenumbers in an infinite fluid-filled cylindrical shell using the asymptotic methods. These expressions are valid for any general circumferential order (n). The shallow shell theory (which is more accurate at higher frequencies) is used to model the cylinder. Initially, the in vacuo shell is dealt with and asymptotic expressions are derived for the shell wavenumbers in the high- and the low-frequency regimes. Next, the fluid-filled shell is considered. Defining a relevant fluid-loading parameter μ, we find solutions for the limiting cases of small and large μ. Wherever relevant, a frequency scaling parameter along with some ingenuity is used to arrive at an elegant asymptotic expression. In all cases, Poisson's ratio ν is used as an expansion variable. The asymptotic results are compared with numerical solutions of the dispersion equation and the dispersion relation obtained by using the more general Donnell-Mushtari shell theory (in vacuo and fluid-filled). A good match is obtained. Hence, the contribution of this work lies in the extension of the existing literature to include arbitrary circumferential orders (n).     

Three-body scattering in configuration space

The asymptotic forms of the wavefunction and Faddeev components in configuration space are shown to determine uniquely the solutions of the Schrödinger or Faddeev differential equations for 2 → (2, 3) and 3 → (2, 3) processes. An antisymmetrized form of the Faddeev differential equation for three equivalent fermions is given and its angular analysis is performed in the general case of local potentials with tensor interaction for neutron-deuteron scattering. We describe a numerical method for solving the corresponding boundary value problem and apply it to scattering and break-up at E_n^1ab = 14.4 MeV in the doublet S state for the four local potentials of Malfliet and Tjon, Reid, de Tourreil and Sprung, and de Tourreil, Rouben and Sprung. For the three realistic potentials, elastic scattering amplitudes differ by 5%, and amplitudes for break-up in the two-neutron state ¹S₀ differ by less than 4%.     

Molecular trajectory algorithm for random walks on percolation systems at criticality in two and three dimensions

Random walk simulations based on a molecular trajectory algorithm are performed on critical percolation clusters. The analysis of corrections to scaling is carried out. It has been found that the fractal dimension of the random walk on the incipient infinite cluster is d_w=2.873±0.008 in two dimensions and 3.78 ± 0.02 in three dimensions. If instead the diffusion is averaged over all clusters at the threshold not subject to the infinite restriction, the corresponding critical exponent k is found to be k=0.3307±0.0014 for two-dimensional space and 0.199 ± 0.002 for three-dimensional space. Moreover, in our simulations the asymptotic behaviors of local critical exponents are reached much earlier than in other numerical methods.     

Asymptotic representations of the period for the nonlinear oscillator

The asymptotic behaviours for small and large amplitudes, A, of the period for a nonlinear oscillator, where the square of the angular frequency depends quadratically on the velocity, are obtained. These asymptotic expressions are compared with the exact period, T(A), and quite an acceptable error for a wide range of amplitudes is obtained. In addition we show that the product of the amplitude and the period, AT(A), reaches 2π when the amplitude tends to infinity.     

Intermittency in branching processes

We study the intermittency properties of two branching processes, one with a uniform and another with a singular splitting kernel. The asymptotic intermittency indices, as well as the leading corrections to the asymptotic linear regime are explicitly computed in an analytic framework. Both models are found to possess a monofractal spectrum with ? _q =q ? 1 and inverse logarithmic corrections. Relations with previous results are discussed.     

Operateurs conjugués et propriétés de propagation

We use an algebraic criteria (based on local positivity of a commutator) which asserts the existence of a direction of propagation for the flowe ^?iHt associated to a self-adjoint operatorH. This criteria is applied to the Hamiltonian of three body quantum systems interacting through long range two body potentials. We found the singular spectral support of the Green functions or equivalently the phase space support of the propagation in the one channel or the coulomb interaction cases. Elementary applications to asymptotic completeness of general three body systems is given in [11b].     

Functional self-similarity,scaling and a renormalization group calculation of the partition function for a non-ideal chain

The hypothesis of asymptotic self-similarity for nonideal polymer chains is used to derive the functional and differential equations of a new renormalization group. These equations are used to calculate the partition functions of randomly jointed chains with hard-sphere excluded-volume interactions. Theoretical predictions are compared with Monte Carlo calculations based on the same microscopic chain model. The excess partition function converges very slowly to its true asymptotic form δQ(N→∞)∼κ^N−1. The conventional asymptotic formula, δQ(N→∞)∼κ^N−1N^γ−1, is found to be applicable for chains of moderate length and for excluded-volume interactions appropriate to the subclass of flexible self-avoiding chains.