An iterative solution of the rough-surface scattering problem

Abstract

An iterative solution to the problem of scattering from a one-dimensional rough surface is obtained for the Dirichlet boundary condition. The advantages of this method are that bounds for convergence of the solution can be established and that the solution may readily be iterated to sufficiently high order in the interaction to examine the rate at which it converges. Absolute convergence of the iterative solution is also a sufficient condition for the convergence of the operator expansion method for surfaces on which the slope is everywhere less than unity. A numerical example of scattering from an echelette grating is considered, and bounds for convergence established. It is found that for scattering from such surfaces the rate at which the iterative solution converges decreases as the surface slope is increased. Corresponding results are found for the operator expansion method.     

An iterative method of the solution of Ising lattice models in a magnetic field

A special re-writing of the sum of δ-functions gives a new graphical representation of thermodynamic quantities. The lowering of the number of points on the reference lattice spin in this representation leads to an iterative scheme by use of which equations for correlation functions and for the order parameter are obtained.     

An exact iterative solution of the atom-surface scattering problem for realistic potentials

A general method, based on high order distorted wave perturbation theory, is developed to obtain exact reflection intensities for low energy atoms scattered by solid surfaces. Resonance shapes are calculated using projection techniques to obtain uniform convergence. Important differences with the hard corrugated wall model arise when the calculations for a corrugated Morse potential are compared with experiment for helium scattered by a Cu(110) surface.     

An approximate solution scheme for the algebraic random eigenvalue problem

A reduced basis formulation is presented for the efficient solution of large-scale algebraic random eigenvalue problems. This formulation aims to improve the accuracy of the first order perturbation method, and also allow the efficient computation of higher order statistical moments of the eigenparameters. In the present method, the two terms of the first order perturbation approximation for the eigenvector are used as basis vectors for Ritz analysis of the governing random eigenvalue problem. This leads to a sequence of reduced order random eigenvalue problems to be solved for each eigenmode of interest. Since, only two basis vectors are used to represent each eigenvector, explicit expressions for the random eigenvalues and eigenvectors can readily be derived. This enables the statistics of the random eigenparameters and the forced response to be efficiently computed. Numerical studies are presented for free and forced vibration analysis of a linear stochastic structural system. It is demonstrated that the reduced basis method gives better results as compared to the first order perturbation method.     

Tunneling resistivity of a one-dimensional random lattice and the petersburg problem

The resistivity of a one-dimensional lattice consisting of randomly distributed conducting and insulating sites is considered. Tunneling resistance of the form ₀ ne ^bn is assumed for a cluster ofn adjacent insulating sites. In the thermodynamic limit, the mean resistance per site diverges at the critical filling fractionp _c =e^–b, while the mean square resistivity fluctuations diverge at the lower filling fraction =p _c ² . Computer simulations of large but finite systems, however, show only a very weak divergence of resistivity atp _c and no divergence of the fluctuations at . For finite lattices, calculation of the resistivity at the critical filling is shown to be simply related to the Petersburg problem. Analytic expressions for the resistivity and resistivity fluctuations are obtained in agreement with the results of computer simulations.     

Exact solution of the O(n) model on a random lattice

We present an exact solution of the O(n) model on a random lattice. The coupling constant space of our model is parametrized in terms of a set of moment variables and the same type of universality with respect to the potential as observed for the one-matrix model is found. In addition we find a large degree of universality with respect to n; namely for n gE ] − 2,2[ the solution can be presented in a form which is valid not only for any potential, but also for any n (not necessarily rational). The cases n = ±2 are treated separately. We give explicit expressions for the genus-zero contribution to the one- and two-loop correlators as well as for the genus-one contribution to the one-loop correlator and the free energy. It is shown how one can obtain from these results any multi-loop correlator and the free energy to any genus and the structure of the higher-genera contributions is described. Furthermore we describe how the calculation of the higher-genera contributions can be pursued in the scaling limit.     

Asymptotic solution of the anti-plane problem for a two-dimensional lattice

Propagation of unsteady waves under the effect of a step point load on a square lattice of springconnected masses is investigated. The problem is solved by two methods. Asymptotic solutions at large time intervals, which describe the behavior of long-wave perturbations, are derived analytically. The solution over the whole time interval for the waves of the entire spectral range is derived by the finite difference method. These solutions are compared, and their good agreement is shown.     

Gauge theory on a random lattice

A general formulation of gauge theory on a random lattice is developed and the strong coupling limit of the Wilson string tension worked out. The confining force found in this strong coupling limit is identical to that predicted by the relativistic string model. In particular, the force between two color-triplet charges is a constant for large separation and the tube of electric flux joining the charges fluctuates, giving it a net thickness proportional to the logarithm of its length.     

Interacting fermions on a random lattice

We extend previous work on the properties of the Dirac lagrangian on two-dimensional random lattices to the case where interaction terms are included. Although for free fermions the chiral symmetry of the doubles is spontaneously broken by their interaction with the lattice and they decouple from long-distance physics, our results in this paper show that all is undone by quantum corrections in an interacting field theory and that the end result is very similar to what is found with Wilson fermions. Two field-theoretical models with interacting fermions are studied by perturbation expansion in the field theory coupling constant. These are a model with one fermion and one boson species interacting via a scalar Yukawa coupling and the massive Thirring model. It is shown that on the random lattice ultraviolet finite diagrams and finite parts of ultraviolet divergent diagrams have the correct continuum limit. Ultraviolet divergent parts can be removed by the same renormalisation procedure as in the continuum, but do not exhibit the same dependence on the lagrangian mass. In the case of the massive Thirring model this causes a fermion mass correction of order the cut-off scale, which breaks the chiral symmetry of the remaining light fermion; there is consequently a fine-tuning problem. In the context of the same model we discuss the effect of the Goldstone boson associated with the spontaneous breakdown of the chiral symmetry of the doubles on two-dimensional models with vector couplings.     

An exact solution on the ferromagnetic face-cubic spin model on a Bethe lattice

The lattice spin model with Q-component discrete spin variables restricted to have orientations orthogonal to the faces of Q  -dimensional hypercube is considered on the Bethe lattice, the recursive graph which contains no cycles. The partition function of the model with dipole–dipole and quadrupole–quadrupole interaction for arbitrary planar graph is presented in terms of double graph expansions. The latter is calculated exactly in case of trees. The system of two recurrent relations (RR) which allows to calculate all thermodynamic characteristics of the model is obtained. The correspondence between thermodynamic phases and different types of fixed points of the RR is established. Using the technique of simple iterations the plots of the zero field magnetization and quadrupolar moment are obtained. Analyzing the regions of stability of different types of fixed points of the system of recurrent relations the phase diagrams of the model are plotted. For Q?2$Q?2$ the phase diagram of the model is found to have three tricritical points, whereas for Q>2$Q>2$ there are one triple and one tricritical points.     

Inhomogeneous random sequential adsorption on a lattice

We study idealized random sequential adsorption on a lattice, with adsorption probabilities inhomogeneous both in space and in time, and including the possibility of cooperativity. Attention is directed to the mean occupancy of a given site as a function of time, which is represented by a weighted random walk on the lattice. In the special case of nearest neighbor exclusion, the walk is transformed to one in which only neighbors of occupied sites can be occupied, but with a renormalized probability. Reduction theorems are presented, with which the general case of a tree lattice is completely solved in inverse form.     

A rapidly convergent iterative solution of the non-LTE line radiation transfer problem

An iterative scheme has been developed for the solution of the non-LTE line radiation transfer problem. The method uses an approximate operator that is deliberately chosen to be local so that it can be easily extended to multidimensional geometry. The difference between the formal and approximate solutions is used as a driving term for the iterations. In one-dimensional, semiinfinite and free-standing slabs, the technique is found to be very fast, robust, and applicable to a large class of problems.     

String interactions on the random lattice

We combine two partons on a random lattice as a vector state. In the ladder approximation, we find that such states have 1/p²$1/p^2$ propagators (after tuning the mass to vanish). We also construct some diagrams which are very similar to 3-string vertices in string field theory for the first oscillator mode. Attaching 3 such lattice states to these vertices, we get Yang–Mills and F³$F^3$ interactions up to 3-point as from bosonic string (field) theory. This gives another view of a gauge field as a bound state in a theory whose only fundamental fields are scalars.     

Numerical simulation of the XY-model on a two-dimensional random lattice

We discuss a numerical simulation of the planar XY-model on a two-dimensional random lattice. Results obtained on the random lattice are compared with those obtained using identical methods on a square lattice, which acts as a “control experiment”. Calculations were made of the average energy per spin, susceptibility per spin and magnetization. The specific heat was obtained by a numerical differentiation of the energy curve. Also, the number of positive (or negative) spin vortices in the system at different temperatures was calculated. Particular attention is paid to the way in which these vortices entered the system, since it is their appearance which signals the phase transition. Numerical results computed at high temperatures are compared with the theoretical values obtained from high-temperature expansions; and those computed at low temperatures, with the results of spin-wave theory. We investigate the effect of varying the weights for the field theory on the random lattice.     

Continuum limit on a 2-dimensional random lattice

We investigate the continuum limit of the Ising model on a 2-dimensional random lattice using Monte Carlo and strong coupling techniques. No evidence is found for deviation from the behaviour of the standard Ising model on a regular lattice; we do not see the modified critical properties found in the bond-diluted Ising model. Our results suggest that the continuum limit is the usual non-interacting Majorana fermion theory.     

Biased random walks on a disordered one-dimensional lattice

We employ the spectral properties of cyclic random matrices to understand the approach to equilibrium of a biased random walk on a disordered lattice. The simplicity of the calculations and the flexibility with which randomness can be modelled makes it a worthwhile alternative.     

Reversible random sequential adsorption on a one-dimensional lattice

We consider the reversible random sequential adsorption of line segments on a one-dimensional lattice. Line segments of length l⩾2 adsorb on the lattice with a adsorption rate K_a, and leave with a desorption rate K_d. We calculate the coverage fraction, and steady-state jamming limits by a Monte Carlo method. We observe that coverage fraction and jamming limits do not follow mean-field results at the large K=K_a/K_d⪢1. Jamming limits decrease when the length of the line segment l increases. However, jamming limits increase monotonically when the parameter K increases. The distribution of two consecutive empty sites is not equivalent to the square of the distribution of isolated empty sites.