A disordered two-dimensional system in a magnetic field: Borel-Padé analysis

We report on a calculation of the density of states and the diffusion constant for the lowest Landau level for an electron moving in two dimensions in a random potential, and under a strong magnetic field. The calculation is based on renormalized perturbation expansion carried out to very high orders, which is summed using the method of Borel-Padé analysis. The results are obtained from an extensive series analysis involving 45 terms in the white noise case and 36 terms in the case of a spatially correlated random potential.     

Independent Component Analysis to enhance performances of Karhunen–Loeve expansions for non-Gaussian stochastic processes: Application to uncertain systems

In the context of engineering systems, an essential step in uncertainty quantification is the development of accurate and efficient representation of the random input parameters. For such input parameters modeled as stochastic processes, Karhunen–Loeve expansion is a classical approach providing efficient representations using a set of uncorrelated, but generally statistically dependent random variables. The dependence structure among these random variables may be difficult to estimate statistically and is thus ignored in many practical applications. This simplifying assumption of independence may lead to considerable errors in estimating the variability in the system state, thus limiting the effectiveness of Karhunen–Loeve expansion in certain cases. In this paper, Independent Component Analysis is exploited to linearly transform the random variables used in Karhunen–Loeve expansion resulting into a set of random variables exhibiting higher order decorrelation. The stochastic wave equation is investigated for numerical illustration whereby the random stiffness coefficient is modeled as a non-Gaussian stochastic process. Under the assumption of independence among the random variables used in the Karhunen–Loeve expansion and Independent Component Analysis representations, the latter provides more accurate statistical characterization of the output process for the specific cases examined.     

The Entropy of a Binary Hidden Markov Process

The entropy of a binary symmetric Hidden Markov Process is calculated as an expansion in the noise parameter ε. We map the problem onto a one-dimensional Ising model in a large field of random signs and calculate the expansion coefficients up to second order in ε. Using a conjecture we extend the calculation to 11th order and discuss the convergence of the resulting series     

A continuum thermal stress theory for crystals based on interatomic potentials

This paper presents a new continuum thermal stress theory for crystals based on interatomic potentials.The effect of finite temperature is taken into account via a harmonic model.An EAM potential for copper is adopted in this paper and verified by computing the effect of the temperature on the specific heat,coefficient of thermal expansion and lattice constant.Then we calculate the elastic constants of copper at finite temperature.The calculation results are in good agreement with experimental data.The thermal stress theory is applied to an anisotropic crystal graphite,in which the Brenner potential is employed.Temperature dependence of the thermodynamic properties,lattice constants and thermal strains for graphite is calculated.The calculation results are also in good agreement with experimental data.     

First passage times for correlated random walks and some generalizations

It is generally difficult to solve Fokker-Planck equations in the presence of absorbing boundaries when both spatial and momentum coordinates appear in the boundary conditions. In this note we analyze a simple, exactly solvable model of the correlated random walk and its continuum analogue. It is shown that one can solve for the moments recursively in one dimension in exact analogy with first passage problems for the Fokker-Planck equation, although the boundary conditions are somewhat more complicated. Further generalizations are suggested to multistate random walks.     

A stochastic study of the non-stationary non-negative random process based on two models of time series: multiplication and addition

As is well-known, there are two contrasting and fundamental models for simulating actual random processes with time series: a multiplication model and an addition model. In this letter, two explicit expressions of the probability density function for a non-stationary non-negative random process (a statistical Laguerre expansion type and a statistical Hermite expansion type) are derived from the above two fundamental viewpoints of modeling a time series, in relation to the statistical method described in a previous paper by the authors, in which the analysis was based on the use of a Hankel transform type characteristic function. The unified theory introduced in this previous paper can be obtained by a very simplified calculation as compared with that of the previous study, by the natural introduction of a random time series multiplication model.     

嵌入随机多项式的抛物方程不确定声场快速算法

为了得到准确且高效计算起伏海洋介质中随机声场的算法,本文将随机多项式展开嵌入到宽角抛物方程声场计算模型(简称RAM模型)中,发展了一种不确定声场的快速算法.其计算结果比使用嵌入随机多项式的窄角抛物方程准确,计算时间小于作为参考的蒙特卡洛方法.在仿真算例中,随机多项式展开法对声强均值、方差、概率密度的计算准确;在一定的随...     

深穿透粒子输运问题的自适应抽样方法

屏蔽计算中的深穿透问题一直是蒙特卡罗计算的一个难题,研究了一种发射点作为驿站的随机游动机制,推导了相应的自适应抽样方法。其主要优势在于,在蒙特卡罗方法求解粒子输运的同时,利用已经获得的信息,自适应地控制各次抽样数,不断完善计算进程。通过对碰撞点引进重要性函数,实现发射点作为驿站的重要性抽样,并结合自适应控制达到最佳抽样状态。数值结果表明:基于发射点作为驿站的自适应抽样方法,在一定程度上克服了深穿透计算中估计值偏低现象。相应的重要函数抽样方法获得了满意的结果。     

Calculation of photoionization cross sections by the absorbing boundary method

A new method for the calculation of quantum state decay rates, the absorbing boundary method (ABM), has recently been developed. In this talk the general form of the ABM will be outlined. The pseudo bound state approach to the calculation of continuum rates such as photoionization cross sections and used by Langhoff and coworkers will be briefly reviewed and shown to lack precision in the absence of information about the line widths and shapes of the pseudo bound states. It will then be noted that the ABM can be used to construct a rigorous new approach to the calculation of photoionization cross sections incorporating the necessary lifetime information for the pseudo bound states while still allowing the computation to be carried out by bound state procedures.     

The treatment of two particles in the continuum in a Green's function theory of the (d,p)-reaction

It is shown that a numerical treatment of two particles in the continuum is possible in applying the Green's function method to the (d, p)-reaction. A basis set for shell model states including the continuum in an approximate way is established and applied to the expansion of the incoming deuteron wave function into these basis states. As a test for the convergence of this expansion the sum of the squares of the expansion coefficients is calculated and found to be 0.965 instead of 1. The main contribution comes from the bound-bound and the bound-continuum configurations. The continuum-continuum contributions can practically be neglected.     

A hybrid particle approach for continuum and rarefied flow simulation

A hybrid particle scheme is presented for the simulation of compressible gas flows involving both continuum regions and rarefied regions with strong translational nonequilibrium. The direct simulation Monte Carlo (DSMC) method is applied in rarefied regions, while remaining portions of the flowfield are simulated using a DSMC-based low diffusion particle method for inviscid flow simulation. The hybrid scheme is suitable for either steady state or unsteady flow problems, and can simulate gas mixtures comprising an arbitrary number of species. Numerical procedures are described for strongly coupled two-way information transfer between continuum and rarefied regions, and additional procedures are outlined for the determination of continuum breakdown. The hybrid scheme is evaluated through a comparison with DSMC simulation results for a Mach 6 flow of N₂ over a cylinder, and good overall agreement is observed. Large potential efficiency gains (over three orders of magnitude) are estimated for the hybrid algorithm relative to DSMC in a simple example involving a rarefied expansion flow through a small nozzle into a vacuum chamber.     

Punktdefekte im kubischen elastischen Kontinuum

The determination of the displacement and strain fields of a point defect in a cubic crystal requires even in the framework of continuum elasticity theory numerical calculation. These fields of elastic dipoles are expanded in suitable vector and tensor fields. The coefficients of this expansion are calculated up to polynomials of 5th and 4th order in the direction cosines using the ratios of elastic constants as parameters. With this expansion the interaction of elastic dipoles in a cubic medium can be calculated. The results have been applied to the interaction of F-centres and of O₂ ^?-centres in alkali halides.     

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.     

A formulation for the characteristic lengths of fcc materials in first strain gradient elasticity via the Sutton–Chen potential

The usual continuum theories are inadequate in predicting the mechanical behavior of solids in the presence of small defects and stress concentrators; it is well known that such continuum methods are unable to detect the change of the size of the inhomogeneities and defects. For these reasons various augmented continuum theories and strain gradient theories have been proposed in the literature. The major difficulty in implication of these theories lies in the lack of information about the additional material constants which appear in such theories. For fcc metals, for the calculation of the associated characteristic lengths which arise in first strain gradient theory, an atomistic approach based on the Sutton–Chen interatomic potential function is proposed. For the validity of the computed characteristic lengths, the phenomenon of the size effect pertinent to a nano-sized circular void within an fcc (111) plane is examined via both first strain gradient theory and lattice statics. Comparison of the results explains the physical ramifications of the characteristic lengths in improving the usual continuum results. Moreover, by reconsideration of the Kelvin problem it is shown that a commonly employed variant of the first strain gradient theory is only valid for a few fcc metals.     

An effective method in calibrating nonlinear errors for phase-shifting interferometry

In phase measurement or digital holography for phase-shifting interferometry, the key role is the variation of reference light wave and recover algorithm based on interferograms and reference phase, so the calculation result is directly affected by phase-shift accuracy. However, because of the errors of nonlinear and other random factors, it is difficult to control the actual phase-shifting amount accurately. In this paper, we aim to propose an efficient method for phase-shifting interferometry which does not require accurate initial estimation of phase-shift amounts, only a few pixels with several randomly shifted interferograms are sufficient for accurate extraction of phase information. This method has reduced the dependence of reference phase, and can obtain phase-shifting amount directly without using complex revised algorithm for correcting phase-shifting nonlinear errors.     

A generalized simple random walk in one dimension related to the Gaussian polynomials

A generalization of the relation between the simple random walk on a regular lattice and the diffusion equation in a continuous space is described. In one dimension we consider a random walk of a walker with exponentially decreasing mobility with respect to time. It has an exact solution of the conditional probability, that is expressed in terms of the Gaussian polynomials, a generalization of binomial coefficients. Taking a suitable continuum limit we obtain the corresponding transport equation from the recursion relation of the discrete random walk process. The kernel of this differential equation is also directly obtained from that conditional probability by the same continuum limit.     

The master two-loop diagram with masses

In every mass case needed for QCD and QED two-point functions, the most difficult two-loop scalar Feynman diagram is reduced, by a systematic dispersive method, to a single integral of logarithms, whose expansion is obtained for large and, when appropriate, small momenta. The new results for the case with an intermediate state comprising three massive particles are needed for the two-loop calculation of fermion propagators.     

Equations of motion for an anisotropic nonlinear elastic continuum in a gravitational field

Equations of motion for an anisotropic nonlinear elastic continuum in the gravitational field are written in a form convenient for numerical calculations. The energy-stress tensor is expressed through scalar and tensor products of three vectors imbedded in the continuum. Examples of expansion of the energy-stress tensor into scalar and tensor invariants corresponding to some crystal classes are given.     

Brownian motion in crystals with topological defects

The diffusion behaviour of a Brownian particle in a crystal with randomly distributed topological defects is analyzed by means of the continuum theory of defects combined with the theory of diffusion on manifolds. A path-integral representation of the diffusion process is used for the calculation of cumulants of the particle position averaged over a defect ensemble. For a random distribution of disclinations the problem of Brownian motion reduces to a known random-drift problem. Depending on the properties of the disclination ensemble, this yields various types of subdiffusional behaviour. In a random array of parallel screw dislocations one finds a normal, but anisotropic, diffusion behaviour of the mean-square displacement. Moreover, the process turns out to be non-Gaussian, and reveals long-time tails in the higher-order cumulants.Dedicated to Professor Herbert Wagner on the occasion of his 60th birthday     

Random Matrices, Graphical Enumeration and the Continuum Limit of Toda Lattices

In this paper we derive analytic characterizations for and explicit evaluations of the coefficients of the matrix integral genus expansion. The expansion itself arises from the large N asymptotic expansion of the logarithm of the partition function of N × N Hermitian random matrices. Its g ^th coefficient is a generating function for graphical enumeration on Riemann surfaces of genus g. The case that we particularly consider is for an underlying measure that differs from the Gaussian weight by a single monomial term of degree 2ν. Our results are based on a hierarchy of recursively solvable differential equations, derived through a novel continuum limit, whose solutions are the coefficients we want to characterize. These equations are interesting in their own right in that their form is related to partitions of 2g + 1 and joint probability distributions for conditioned random walks. K. D. T-R McLaughlin was supported in part by NSF grants DMS-0451495 and DMS-0200749, as well as a NATO Collaborative Linkage Grant “Orthogonal Polynomials: Theory, Applications, and Generalizations” Ref no. PST.CLG.979738. N. M. Ercolani and V. U. Pierce were supported in part by NSF grants DMS-0073087 and DMS-0412310.