Power-Law Distributions for the Free Path Length in Lorentz Gases

It is well known that, in the Boltzmann–Grad limit, the distribution of the free path length in the Lorentz gas with disordered scatterer configuration has an exponential density. If, on the other hand, the scatterers are located at the vertices of a Euclidean lattice, the density has a power-law tail proportional to $\xi ^{-3}$ . In the present paper we construct scatterer configurations whose free path lengths have a distribution with tail $\xi ^{-N-2}$ for any positive integer $N$ . We also discuss the properties of the random flight process that describes the Lorentz gas in the Boltzmann–Grad limit. The convergence of the distribution of the free path length follows from equidistribution of large spheres in products of certain homogeneous spaces, which in turn is a consequence of Ratner’s measure classification theorem.     

Derivation of the Boltzmann equation for a Fermi gas

Point scatterers are placed on the real line such that the distances between scatterers are independent identically distributed random variables (stationary renewal process). For a fixed configuration of scatterers a particle performs the following random walk: The particle starts at the pointx with velocityυ, ¦υ¦=1. In between scatterers the particle moves freely. At a scatterer the particle is either transmitted or reflected, both with probability 1/2. For given initial conditions of the particle the velocity autocorrelation function is a random variable on the scatterer configurations. If this variable is averaged over the distribution of scatterers, it decays not faster thant ^?3/2.     

A finite-time exponent for random Ehrenfest gas

We consider the motion of a system of free particles moving on a plane with regular hard polygonal scatterers arranged in a random manner. Calling this the Ehrenfest gas, which is known to have a zero Lyapunov exponent, we propose a finite-time exponent to characterize its dynamics. As the number of sides of the polygon goes to infinity, when polygon tends to a circle, we recover the usual Lyapunov exponent for the Lorentz gas from the exponent proposed here. To obtain this result, we generalize the reflection law of a beam of rays incident on a polygonal scatterer in a way that the formula for the circular scatterer is recovered in the limit of infinite number of vertices. Thus, chaos emerges from pseudochaos in an appropriate limit.     

Thermodynamic limit for the one-dimensional Ising systems with random interaction of finite range

The existence of the thermodynamic limit is proved for the random one-dimensional Ising systems under the assumption that the interaction energies are random variables taking on continuous values and the distribution of the random variables is given by a continuous function. It is assumed that the total number of possible configurations for each lattice site is finite and the range of interaction is finite.     

Free Path Lengths in Quasi Crystals

The Lorentz gas is a model for a cloud of point particles (electrons) in a distribution of scatterers in space. The scatterers are often assumed to be spherical with a fixed diameter d, and the point particles move with constant velocity between the scatterers, and are specularly reflected when hitting a scatterer. There is no interaction between point particles. An interesting question concerns the distribution of free path lengths, i.e. the distance a point particle moves between the scattering events, and how this distribution scales with scatterer diameter, scatterer density and the distribution of the scatterers. It is by now well known that in the so-called Boltzmann–Grad limit, a Poisson distribution of scatterers leads to an exponential distribution of free path lengths, whereas if the scatterer distribution is periodic, the free path length distribution asymptotically behaves as a power law.     

Effective scatterer approximation for mean field modeling with application to vegetation

A new electromagnetic method is developed for the mean field analysis of discrete random media in which particles are distributed in disjoint regions in space. This type of distribution can be observed in organized vegetation canopies, photonic band gap materials and disordered crystals. The method is based on an ‘effective scatterer’ concept. As far as the mean field is concerned, the random problem is approximated by a deterministic one composed of effective scatterers. Application of the effective scatterer approach to two-dimensional mean field propagation through a cylinder distribution that is periodic on average is verified by Monte Carlo simulations. Comparisons with other known methods such as the Foldy approximation are also presented.     

Stochastic effects on the bistatic transfer function of a planar scatterer distribution

ABSTRACT

We evaluate the effects of several stochastic factors on signal transmission through a planar distribution of stationary scatterers, with non-collocated transmitter and receiver (bistatic configuration). The transmission channel is described by means of its transfer function, which –as a result of randomness in the scatterer distribution– fluctuates around its expectation value. Specifically, we consider randomness in the scatterer positions (both on the plane and in height), in their radar cross sections, and in the scattering phases. Our analytical results provide a quantitative relation between the parameters that characterize random components of each kind, and the fluctuations that alter the transfer function.     

基于Wigner-Ville分布用超声回波测量随机介质的特征

生物组织微结构具有随机介质的特性,文中构建了生物组织超声散射的一维随机介质模型,提出用Ｗｉｇｎｅｒ－Ｖｉｌｌｅ分布函数方法估计该随机仙擀的空间分布特征,仿真和实验结果表明这一方法具有良好的空间分辨率,能较好地检测出随机介质微小散射元的个数和空间分布同时能精确地保估出散射元平均间距,研究结果表明该方法还具有良好的抗噪声性等优良特性。     

A stochastic discrete model of tissue and its ultrasonic backscattering coefficient

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Statistics of two-dimensional point vortices and high-energy vortex states

Results from the theory ofU-statistics are used to characterize the microcanonical partition function of theN-vortex system in a rectangular region for largeN, under various boundary conditions, and for neutral, asymptotically neutral, and nonneutral systems. Numerical estimates show that the limiting distribution is well matched in the region of major probability forN larger than 20. Implications for the thermodynamic limit are discussed. Vortex clustering is quantitatively studied via the average interaction energy between negative and positive vortices. Vortex states for which clustering is generic (in a statistical sense) are shown to result from two modeling processes: the approximation of a continuous inviscid fluid by point vortex configurations; and the modeling of the evolution of a continuous fluid at high Reynolds number by point vortex configurations, with the viscosity represented by the annihilation of close positive-negative vortex pairs. This last process, with the vortex dynamics replaced by a random walk, reproduces quite well the late-time features seen in spectral integration of the 2d Navier-Stokes equation.     

Speckle characteristics of simulated deep Fresnel region under shadowing effect

After the three-dimensional self-affine fractal random surface simulation, we use the optical scattering theory to calculate the deep Fresnel region speckle(DFRS) under consideration of the more strict shadowing effect. The evolution of DFRS with the scattering distance and the intensity probability distribution are studied. It is found that the morphology of the scatterer has an antisymmetric relationship with the intensity distribution of DFRS, and the effect of micro-lenses on the scattering surface causes the intensity probability distribution of DFRS to deviate from the Gaussian speckle in the high light intensity area.     

Scattering of light from particles with semisoft boundaries

A three-dimensional multi-Gaussian function, being a finite sum of Gaussian functions, is adopted for modeling of a spherically symmetric scatterer with a semisoft boundary, i.e. such that has continuous and adjustable drop in the index of refraction. A Gaussian sphere and a hard sphere are the two limiting cases when the number of terms in multi-Gaussian distribution is one and infinity, respectively. The effect of the boundary's softness on the intensity distribution of the scattered wave is revealed. The generalization of the model to random scatterers with semisoft boundaries is also outlined.     

Double passage analysis in random media using two-scale random propagators

Abstract

In location and remote sensing experiments there arise a number of effects related to the double passage of the backscattered field through the same random inhomogeneities as the incident one. To account for the correlation of the forward–backward propagating events, there is a need for a measure in which the random information along the propagation path is preserved. For the generation of even statistical moments, the relevant measure defined in the recently formulated stochastic geometrical theory of diflraction is the two-point random function (TPRF)—a paired field measure which is propagated along the geometrical rays of the deterministic background medium. From this function all even statistical moments can be generated. Here we present an approximate analytical solution for the high-frequency propagator obtained by applying the multiscale expansion asymptotic procedure to the partial differential equation governing the propagation a1 the TPRF. The test of the solution is performed on canonical backscattering problems based on point source–point scatterer and paint source–plane mirror configurations, which justifies its further application for construction of the coherence measures of the rctrareflected field. Coherence properties of the plane and spherical wavefields reflected backward by a plane mirror were investigated. Further, we investigated the intensity enhancement effects observed in the double passage of a Gaussian beam retroreflected from a plane mirror. Asymptotic expressions lor the retroreflected intensity are obtained, and their computations show good agreement with the direct numerical evaluations.     

Vicious random walkers and a discretization of Gaussian random matrix ensembles

The vicious random walker problem on a one-dimensional lattice is considered. Many walkers take simultaneous steps on the lattice and the configurations in which two of them arrive at the same site are prohibited. It is known that the probability distribution of N walkers after M steps can be written in a determinant form. Using an integration technique borrowed from the theory of random matrices, we show that arbitrary kth order correlation functions of the walkers can be expressed as quaternion determinants whose elements are compactly expressed in terms of symmetric Hahn polynomials.     

随机介质中的矢量辐射传输理论

本文讨论了极化电磁波在随机介质中多次散射,传输和热辐射的斯托克斯矢量的辐射传输理论。其中包括随机分布离散的球形和非球形粒子的矢量辐射传输方程,离散坐标-特征分析法,付利叶变换,迭代法等数值解。讨论了非球形粒子的穆勒矩阵。并研究了密集分布的散射粒子介质的辐射传输理论,考虑了密集粒子散射的相干性,计算了有效传播常数。理论及数值结果与实验作了很好的比较。     

系统可靠度置信下限估计方法及其应用

针对寿命服从指数分布的随机截尾样本，研究单元及系统可靠度的评估方法，给出了系统可靠度置信下限估计方法，该方法同样适用于工程中常见的定时截尾样本与定数截尾样本的系统可靠度评估问题。首先通过分位数填充算法将随机截尾数据补充成虚拟完全样本，然后基于信仰推断，给出系统内每个单元失效率的信仰分布，最后根据指数分布的特点，从失效率角度出发，采用蒙特卡罗法对单元失效率进行随机抽样，结合系统可靠性模型得到系统失效率的分布，在给定置信水平下即可得到系统失效率上限，进而得到工程上特别关注的系统可靠度置信下限。     

Localization and Propagation in Random Lattices

We analyze the motion of a particle on random lattices. Scatterers of two different types are independently distributed among the vertices of such a lattice. A particle hops from a vertex to one of its neighboring vertices. The choice of neighbor is completely determined by the type of scatterer at the current vertex. It is shown that on Poisson and vectorizable random triangular lattices the particle will either propagate along some unbounded strip or be trapped inside a closed strip. We also characterize the structure of a localization zone contained within a closed strip. Another result shows that for a general class of random lattices the orbit of a particle will be bounded with probability one.     

Method of improved scatterer size estimation and application to parametric imaging using ultrasound

The frequency dependence of RF signals backscattered from random media (tissues) has been used to describe the microstructure of the media. The frequency dependence of the backscattered RF signal is seen in the power spectrum. Estimates of scatterer properties (average scatterer size) from an interrogated medium are made by minimizing the average squared deviation (MASD) between the measured power spectrum and a theoretical power spectrum over an analysis bandwidth. Estimates of the scatterer properties become increasingly inaccurate as the average signal to noise ratio (SNR) over the analysis bandwidth becomes smaller. Some frequency components in the analysis bandwidth of the measured power spectrum will have smaller SNR than other frequency components. The accuracy of estimates can be improved by weighting the frequency components that have the smallest SNR less than the frequencies with the largest SNR in the MASD. A weighting function is devised that minimizes the noise effects on the estimates of the average scatterer sizes. Simulations and phantom experiments are conducted that show the weighting function gives improved estimates in an attenuating medium. The weighting function is applied to parametric images using scatterer size estimates of a rat that had developed a spontaneous mammary tumor.     

Time reversal of speckle noise

Focusing a wave in an unknown inhomogeneous medium is an open problem in wave physics. This work presents an iterative method able to focus in pulse-echo mode in an inhomogeneous medium containing a random distribution of scatterers. By performing a coherent summation of the random echoes backscattered from a set of points surrounding the desired focus, a virtual bright pointlike reflector is generated. A time-reversal method enables an iterative convergence towards the optimal wave field focusing at the location of this virtual scatterer. Thanks to this iterative time-reversal process, it is possible to focus at any arbitrary point in the heterogeneous medium even in the absence of pointlike source. An experimental demonstration is given for the correction of strongly distorted images in the field of medical ultrasound imaging. This concept enables envisioning many other applications in wave physics.