Application of Synthetic Kernel (SKN) method to participating linearly anisotropically scattering solid spherical medium

The Synthetic Kernel (SK_N) method is applied to a solid spherical absorbing, emitting and linearly anisotropically scattering homogeneous and inhomogeneous medium. The SK_N method relies on approximating the integral transfer kernels by Synthetic Kernels. The radiative integral transfer equation is then reducible to a set of coupled second-order differential equations. The SK_N method, which uses Gauss quadratures, is tested against integral equation and the discrete-ordinates S₈ solutions for various optical radius and scattering albedo variations.     

A fast directional algorithm for high-frequency electromagnetic scattering

This paper is concerned with the fast solution of high-frequency electromagnetic scattering problems using the boundary integral formulation. We extend the O(N log N) directional multilevel algorithm previously proposed for the acoustic scattering case to the vector electromagnetic case. We also detail how to incorporate the curl operator of the magnetic field integral equation into the algorithm. When combined with a standard iterative method, this results in an almost linear complexity solver for the combined field integral equations. In addition, the butterfly algorithm is utilized to compute the far field pattern and radar cross section with O(N log N) complexity.     

The quantum dual string wave functional in Yang-Mills theories

From any solution of the classical Yang-Mills equations, we define a string wave functional based on the Wilson loop integral. Its precise definition is given by replacing the string by a finite set of N points, and taking the limit N → ∞. We show that this functional satisfies the Schrödinger equation of the relativistic dual string to leading order in N. We speculate about the relevance of this object to the quantum problem.     

A simple reduction procedure for the three- and N-body scattering equations

A generalized form of the two-body Kowalski-Noyes method is shown to provide a both simple and powerful unitary reduction of the three- and N-body scattering equations. Employing generalized half-off-shell functions that satisfy of-sshell but real and non-singular integral equations, the reduction directly leads to on-shell integral equations for the scattering amplitudes. Physically, it is simple example of how the scattering problem can be split into an internal and an external part.     

N-body saddle-point approximation

By applying the saddle-point approximation to the N-body Feynman path integral formulation, the classical Hartree-Fock Molecular Orbital (M.O.) equations of quantum chemistry are obtained.     

Supersymmetry in stochastic processes with higher-order time derivatives

A supersymmetric path-integral representation is developed for stochastic processes whose Langevin equation contains any number N of time derivatives, thus generalizing the presently available treatment of first-order Langevin equations by Parisi and Sourlas [Phys. Rev. Lett. 43 (1979) 744; Nucl. Phys. B 206 (1982) 321] to systems with inertia (Kramers' process) and beyond. The supersymmetric action contains N fermion fields with first-order time derivatives whose path integral is evaluated for fermionless asymptotic states.     

The EPS method: A new method for constructing pseudospectral derivative operators

We develop a new type of derivative matrix for pseudospectral methods. The norm of these matrices grows at the optimal rate O(N²) for N-by-N matrices, in contrast to standard pseudospectral constructions that result in O(N⁴) growth of the norm. The smaller norm has a big advantage when using the derivative matrix for solving time dependent problems such as wave propagation. The construction is based on representing the derivative operator as an integral kernel, and does not rely on the interpolating polynomials. In particular, we construct second derivative matrices that incorporate Dirichlet or Neumann boundary conditions on an interval and on the disk, but the method can be used to construct a wide variety of commonly used operators for solving PDEs and integral equations. The construction can be used with any quadrature, including traditional Gauss–Legendre quadratures, but we have found that by using quadratures based on prolate spheroidal wave functions, we can achieve a near optimal sampling rate close to two points per wavelength, even for non-periodic problems. We provide numerical results for the new construction and demonstrate that the construction achieves similar or better accuracy than traditional pseudospectral derivative matrices, while resulting in a norm that is orders of magnitude smaller than the standard construction. To demonstrate the advantage of the new construction, we apply the method for solving the wave equation in constant and discontinuous media and for solving PDEs on the unit disk. We also present two compression algorithms for applying the derivative matrices in O(N log N) operations.     

Unitary approach to quasielastic scattering reactions

The possibility of deriving an approximate unitary solution to integral Faddeev equations within the K-matrix formalism is considered. Explicit expressions for the amplitudes of elastic, inelastic, and quasielastic three-body scattering are obtained under the assumption of a mechanism of a truly single collision. Specific calculations are performed for quasielastic-scattering reactions of the d(N, 2N)N type. Good agreement between the results of these calculations and experimental data indicates that, in developing approximate methods, it is highly desirable to respect fundamental physical principles.     

From crystal steps to continuum laws: Behavior near large facets in one dimension

The passage from discrete schemes for surface line defects (steps) to nonlinear macroscopic laws for crystals is studied via formal asymptotics in one space dimension. Our goal is to illustrate by explicit computations the emergence from step motion laws of continuum-scale power series expansions for the slope near the edges of large, flat surface regions (facets). We consider surface diffusion kinetics via the Burton, Cabrera and Frank (BCF) model by which adsorbed atoms diffuse on terraces and attach-detach at steps. Nearest-neighbor step interactions are included. The setting is a monotone train of N steps separating two semi-infinite facets at fixed heights. We show how boundary conditions for the continuum slope and flux, and expansions in the height variable near facets, may emerge from the algebraic structure of discrete schemes as N→∞. Our technique relies on the use of self-similar discrete slopes, conversion of discrete schemes to sum equations, and their reduction to nonlinear integral equations for the continuum-scale slope. Approximate solutions to the continuum equations near facet edges are constructed formally by direct iterations. For elastic-dipole and multipole step interactions, the continuum slope is found in agreement with a previous hypothesis of ‘local equilibrium’.     

Generic van der Waals equation of state for polymers, modified free volume theory, and the self-diffusion coefficient of polymeric liquids

In this paper, a molecular theory of self-diffusion coefficient is developed for polymeric liquids (melts) on the basis of the integral equation theory for site-site pair correlation functions, the generic van der Waals equation of state, and the modified free volume theory of diffusion. The integral equations supply the pair correlation functions necessary for the generic van der Waals equation of state, which in turn makes it possible to calculate the self-diffusion coefficient on the basis of the modified free volume theory of diffusion. A random distribution is assumed for minimum free volumes for monomers along the chain in the melt. More specifically, a stretched exponential is taken for the distribution function. If the exponents of the distribution function for minimum free volumes for monomers are chosen suitably for linear polymer melts of N monomers, the N dependence of the self-diffusion coefficient is N⁻¹ for the small values of N, an exponent predicted by the Rouse theory, whereas in the range of 2.3?lnN?4.5 the N dependence smoothly crosses over to N⁻², which is reminiscent of the exponent by the reptation theory. However, for lnN?4.5 the N dependence of the self-diffusion coefficient differs from N⁻², but gives an N dependence, N^−2−δ(0<δ<1), consistent with experiment on polymer melts in the range. For polyethylene δ≈0.48 for the parameters chosen for the stretched exponential. Because the stretched exponential function contains undetermined parameters, the N dependence of diffusion becomes semiempirical, but once the parameters are chosen such that the N dependence of D can be successfully given for a polymer melt, the temperature dependence of the self-diffusion coefficient can be well predicted in comparison with experiment. The theory is satisfactorily tested against experimental and simulation data on the temperature dependence of D for polyethylene and polystyrene melts.     

Efficient numerical methods for multiple surfactant-coated bubbles in a two-dimensional stokes flow

We present efficient and highly accurate numerical methods to compute the deformation of surfactant-coated, two-dimensional bubbles in a slow viscous flow. Surfactant acts to locally alter the surface tension and thereby change the nature of the interface motion. In this paper, we restrict our attention to the case of a dilute insoluble surfactant. The convection–diffusion equation for the surfactant concentration on the interface is coupled with the Stokes equations in the fluid domain through a boundary condition based on the Laplace-Young condition. The Stokes equations are first recast as an integral equation and then solved using a fast-multipole accelerated iterative procedure. The computational cost per time-step is only O(N log N) operations, with N being the number of discretization points on the interface. The bubble interfaces are described by a spectral mesh and is advected according to the fluid velocity in such a manner so as to preserve equal arc length spacing of marker points. This equal arc length framework has the dual advantage of dynamically maintaining the spatial mesh and allowing efficient, implicit treatment of the stiffest terms in the dynamics. Several phenomenologically different examples are presented.     

The multiple scattering and N-body approaches to nuclear reactions

The relationship between conventional multiple scattering approaches and the recently developed N-body approaches to nuclear reactions is considered with a view towards elastic scattering applications. Connectivity expansions in the N-body approach and multiple scattering expansions in the Watson approach are developed by a common technique so that a comparison of the physical content of each can be made. In the N-body case this leads to a new derivation of the equations of Bencze, Redish, and Sloan in both particle-labelled and partition-labelled form and this yields new insight into the minimal dimensionality of these equations and into the role of channel coupling schemes within this formulation. The relative simplicity and generality with which these results are obtained is designed to be easily understood by those unfamiliar with N-body formalisms. The two approaches are contrasted first for the three-particle problem and subsequently for the many-body problem. We argue that a strict adherence to the connected-kernel property which is advantageous for the three-particle problem may not be so advantageous for the many-body elastic scattering problem. Undesirable physical characteristics of the connectivity expansion for elastic scattering are identified and their rectification is discussed. The off-shell transformation associated with the N-body approach is examined critically. The origin of the multiplicity of N-body coupling schemes is elucidated. It is shown that a modified concept of connectivity, called inclusive connectivity, can be introduced to guide expansions which can be truncated in a physically meaningful way. The inclusive connectivity expansion is seen to be identical to the spectator expansion for an elementary projectile but differs in the case of a composite projectile. Extant elastic scattering optical potential formulations based on the two concepts of connectivity are compared and contrasted. We show that connected kernel integral equations of the few-body type are required for computation of the individual low-order terms of the inclusive connectivity expansion of the optical potential.     

Exact thermodynamics of the Hubbard chain: free energy and correlation lengths

We study the one-dimensional Hubbard model at finite temperatures in the quantum transfer matrix approach. The eigenvalue equations of this matrix are obtained by a nested Bethe ansatz. The largest and next-largest eigenvalues yield the free energy as well as the correlation lengths of the system. An equivalent set of four integral equations is derived from the Bethe ansatz equations. The limit of Trotter-Suzuki number N → ∞ is taken analytically. For half-filling the final equations are studied in the low-temperature limit yielding analytic expressions for the free energy and spin-spin correlation length. Numerical results are presented for intermediate temperatures.     

Fast integral equation methods for the modified Helmholtz equation

We present integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(x) ? α²Δu(x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of our methods on several numerical examples, and we show that they have both the ability to handle highly complex geometry and the potential to solve large-scale problems.     

A multilevel Cartesian non-uniform grid time domain algorithm

A multilevel Cartesian non-uniform grid time domain algorithm (CNGTDA) is introduced to rapidly compute transient wave fields radiated by time dependent three-dimensional source constellations. CNGTDA leverages the observation that transient wave fields generated by temporally bandlimited and spatially confined source constellations can be recovered via interpolation from appropriately delay- and amplitude-compensated field samples. This property is used in conjunction with a multilevel scheme, in which the computational domain is hierarchically decomposed into subdomains with sparse non-uniform grids used to obtain the fields. For both surface and volumetric source distributions, the computational cost of CNGTDA to compute the transient field at N_s observation locations from N_s collocated sources for N_t discrete time instances scales as O(N_tN_slogN_s) and O(N_tN_slog²N_s) in the low- and high-frequency regimes, respectively. Coupled with marching-on-in-time (MOT) time domain integral equations, CNGTDA can facilitate efficient analysis of large scale time domain electromagnetic and acoustic problems.     

Dynamical connections between quark fragmentation functions

Assuming that the quark kicked by a virtual photon emits pions one by one, integral equations connecting the fragmentation functions D_u^π+ (z) and D_u^π? (z) are obtained. It is shown that they have a plateau the height of which can be determined from the multiplicity difference N_p^π+ (z) ? N_p^π? (z). Comparison with experiment is made.     

Large N classical equations and their quantum significance

We show that the large N limits of a wide variety of vector models may be obtained by studying the classical equations of motion. In particular, we derive a constraint which allows us to choose solutions of the classical field equations which directly give the correlation functions of N → ∞ quantum system. Models studied here include quantum mechanics on a sphere, two-dimensional linear and nonlinear O(N) field theories and the CP^N model.     

Integral equations satisfied by the H- X- and Y-functions and the albedo problem

We come back to a nonlinear integral equation satisfied by the function H, which is distinct from the classical H-equation. Established for the first time by Busbridge (1955), it appeared occasionally in the literature since then. First of all, this equation is generalized over the whole complex plane using the method of residues. Then its counterpart in a finite slab is derived; it consists in two series of integral equations for the X- and Y-functions. These integral equations are finally applied to the solution of the albedo problem in a slab.     

Classical behavior of large N fermionic systems

The large N limit of several fermionic systems in two dimensions is shown to be obtainable by doing classical mechanics. This generalizes results previously derived for bosonic models. For these types of theories, the reduction to a classical system of equations is closely related to the path integral quantization scheme of Dashen. Hasslacher and Neveu. Using this relationship, we are able to gain further insight into the workings of both approaches.