Random Walks and Polymers in the Presence of Quenched Disorder

After a general introduction to the field, we describe some recent results concerning disorder effects on both ‘random walk models’, where the random walk is a dynamical process generated by local transition rules, and on ‘polymer models’, where each random walk trajectory representing the configuration of a polymer chain is associated to a global Boltzmann weight. For random walk models, we explain, on the specific examples of the Sinai model and of the trap model, how disorder induces anomalous diffusion, aging behaviours and Golosov localization, and how these properties can be understood via a strong disorder renormalization approach. For polymer models, we discuss the critical properties of various delocalization transitions involving random polymers. We first summarize some recent progresses in the general theory of random critical points: thermodynamic observables are not self-averaging at criticality whenever disorder is relevant, and this lack of self-averaging is directly related to the probability distribution of pseudo-critical temperatures T _c(i,L) over the ensemble of samples (i) of size L. We describe the results of this analysis for the bidimensional wetting and for the Poland–Scheraga model of DNA denaturation.Conference Proceedings “Mathematics and Physics”, I.H.E.S., France, November 2005     

Interband transitions and electronic Raman continuum in systems with strong inelastic scattering: Applications to YBa2Cu3O7-δ

We consider a model for the electronic Raman continuum which takes into account strong inelastic scattering and interband transitions. Calculations are based on four-vertex Raman scattering diagrams (Kawabata formalism) within the RPA for Coulomb interaction and the ladder diagram Bethe-Salpeter equation for the vertex. We apply this method to an analysis of the nature of the electronic Raman continuum in the normal state of the high-T _c superconductor YBa₂Cu₃O₇. In numerical calculations we take into account all the self-energy effects and make simulations for vertex corrections assuming that inelastic scattering is due to electron-phonon interaction. Theab-plane polarized continuum contains a large contribution from interband processes and does not depend strongly on temperature and inelastic scattering strength. The in-plane anisotropy is determined by interband transitions rather than by anisotropy of the Fermi surface. The ZZ continuum contains only weak contribution from interband transitions. It can be crudely described within a single band model with inelastic scattering and is strongly dependent on the relaxation rates of inelastic scattering. The nature of the oxygen-deficiency dependence of the Raman spectra is also commented upon.     

Mott Law as Lower Bound for a Random Walk in a Random Environment

We consider a random walk on the support of an ergodic stationary simple point process on ℝ^d, d≥2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.     

Mott Law as Upper Bound for a Random Walk in a Random Environment

We consider a random walk on the support of an ergodic simple point process on , d ≥ 2, furnished with independent energy marks. The jump rates of the random walk decay exponentially in the jump length and depend on the energy marks via a Boltzmann–type factor. This is an effective model for the phonon–induced hopping of electrons in disordered solids in the regime of strong Anderson localization. Under some technical assumption on the point process we prove an upper bound for the diffusion matrix of the random walk in agreement with Mott law. A lower bound for d ≥ 2 in agreement with Mott law was proved in [8].     

Coherent effects of multiple scattering for scalar and electromagnetic fields: Monte-Carlo simulation and Milne-like solutions

Based on an expansion of the Bethe-Salpeter equation in scattering orders a semi-analytic approach for simulation of coherent phenomena of multiple scattering in random media has been developed. We found that for scalar field the manifestation of these phenomena, observed as temporal field correlation function and coherent backscattering, are universal and well agreed with the results predicted by diffusion approximation. For the electromagnetic field the temporal correlation function and coherent backscattering are noticeably differ from those found for the scalar field, depending strongly on the scattering anisotropy. The obtained numerical results, for the first time to our knowledge, are compared directly with the known generalizations of the Milne solution.     

Simulation of coherent backscattering of light in nematic liquid crystals

Multiple scattering of light by the fluctuations of the director in a nematic liquid crystal (NLC) aligned by a magnetic field is considered. A peak of coherent backscattering is calculated by numerical simulation. Since the indicatrix of single scattering for a liquid crystal (LC) is known exactly, the calculations are carried out without any simplifying assumptions on the parameters of the liquid crystal. Multiple scattering is simulated as a random walk of photons in the medium. A peak of coherent backscattering in such a medium is very narrow; therefore, the so-called semianalytical method is applied. The parameters of the backscattering peak obtained by numerical simulation are compared with the available experimental data and with the results of analytical approximations. It turns out that the experimental data are in good agreement with the results of simulation. The results of numerical simulation adequately describe the anisotropy and the width of the backscattering peak.     

Application of a coherent model in simulating the backscattering coefficient of a mangrove forest

Abstract

In this paper, a single scattering model is presented for a coherent forest scattering simulation. It is tested on the backscattering coefficient of mangrove forests, which are known to involve large coherent effects. Analysis of branches, leaves and ground contributions is done to understand the backscattering coefficient composition. Finally the sensitivity of the code is investigated.     

Coherent backscattering of electromagnetic waves in a magnetised plasma

Summary A microwave coherent backscattering experiment has been carried out on Mirabelle, a weakly ionised plasma device, with the objective of measuring the electron density fluctuation level. The experiment is a preliminary step in order to prepare the detection system for a microwave stimulated backscattering experiment. The incident electromagnetic wave is focused in front of a plane grid which excites ion acoustic or electron Bernstein waves inducing fluctuations in the plasma. The backscattering signal is collected by the launching circuit and detected by homodyne mixing. The typical ratio of the scattered power to the incident power is about 10⁻¹² and the relative density fluctuations are of the order of δn _e/n _e=10⁻³ against a background electron density ofn _e=1–5·10⁹ cm⁻³. The backscattering measurement is compared with Langmuir probe measurements. The spectral width of the backscattered signal has also been studied, by taking into account effects due to the incident wave focusing and plasma wave damping. The authors of this paper have agreed to not receive the proofs for correction     

Deducing residual self-interference of inelastic photons in coherent backscattering of intense laser light from two atoms

A coherent backscattering of intense laser light by two isotropic atoms is studied in the helicity preserving polarization channel. It is demonstrated that single scattering has a non-negligible contribution to the background intensity L(θ) at small angles θ with respect to the backwards direction. This contribution can be subtracted from the total signal, and the value of L(0) necessary for evaluating the coherent backscattering enhancement factor – inferred from measurements of the backscattered light intensity beyond the interference peak.     

On the “supercollimation” of X-ray beams in rough interfacial channels

The transmission of x rays through rough submicron narrow channels is investigated by numerical simulation with diffraction and decay of coherence taken into account. It is found that transmission is strongly increased for directions within the diffraction limit λ/d (d is the channel width). For larger angles strong roughness scattering results in rapid decay of coherence and absorption of the x-ray beams. When the coherent part is a significant portion of the transmitted beam, its divergence is also within the diffraction limit, which can be an order of magnitude smaller than the Fresnel angle of total external reflection. The effects are explained with the statistical theory of x-ray scattering in a rough transitional layer. Such “supercollimation” can be used for fine angular discrimination of x radiation and for the production of very narrow diffraction-quality x-ray beams. Zh. éksp. Teor. Fiz. 69, No. 10, 686–690 (25 May 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Effects of nondiffusive wave propagation upon coherent backscattering by turbid media

The nondiffusive contribution to the coherent backscattering intensity is calculated for the media with relatively large particles (size a is greater than wavelength λ). The results are in good agreement with the experimental data at the wings of the angular spectrum of the coherent backscattering. The shape of the backscattering peak is analyzed for strongly absorbing media. The correlation function of the intensity fluctuations is calculated for the scattering by Brownian particles at relatively large time shifts.     

Geometrical implications of the compound representation for weakly scattered amplitudes

It is known that a random walk model yields the multiplicative representation of a coherent scattered amplitude in terms of a complex Ornstein–Uhlenbeck process modulated by the square root of the cross-section. A corresponding biased random walk enables the derivation of the dynamics of a weak coherent scattered amplitude as a stochastic process in the complex plane. Strong and weak scattering patterns differ regarding the correlation structure of their radial and angular fluctuations. Investigating these geometric characteristics yields two distinct procedures to infer the scattering cross-section from the phase and intensity fluctuations of the scattered amplitude. These inference techniques generalize an earlier result demonstrated in the strong scattering case. Their significance for experimental applications, where the cross-section enables tracking of anomalies, is discussed.     

Weak localization effects in the second-harmonic light scattered by random systems of particles

We report rigorous numerical simulations that show the presence of coherent backscattering effects in the second-harmonic generation and scattering of light by random systems of two-dimensional particles. Since the medium composing the particles is assumed to be homogeneous and isotropic, the second-harmonic field is generated mainly by surface effects. For the fundamental frequency, the results present a clear enhanced backscattering peak. In contrast, the second-harmonic scattering patterns present an intensity dip in the backscattering direction.     

Coherent backscattering in nematic liquid crystals in an external magnetic field

We consider multiple light scattering in a nematic liquid crystal. Using the Monte Carlo method, we calculate for the first time the effect of a magnetic field on the shape of the peak of coherent backscattering taking into account the long-range action of fluctuations of the orientational order and anisotropy of the scattering length. For a small number of initial and final scattering events, we take into account the ordinary mode of light, which is weakly scattered in a nematic liquid crystal (NLC), whereas a strongly scattered extraordinary mode is taken into account for all scattering events. For simplicity, we use a single-constant approximation of the NLC elastic moduli. We show that the angular shape of the peak of coherent backscattering remains nearly unchanged, whereas the magnetic field and the scattering phase function vary by several orders of magnitude.     

Nonlinear conductance fluctuations in a mesoscopic ring

We study the conductance of a single particle on a ring subject to an arbitrary dc electric field, which is generated by a linearly in time increasing magnetic flux. The full quantum mechanical time development is calculated numerically by splitting the dynamics into independent consecutive Zener tunneling transitions and free motion on the ring. The Zener transitions occur near the avoided crossings of the bandstructure which arises from the adiabatic eigenstates as a function of flux in the presence of a static scattering potential. To account for the necessary dissipation the particle is coupled to an appropriate oscillator bath which is adjusted to give a strictly linear current-voltage characteristic for arbitrary voltage and temperature in the absence of scattering. Taking a single δ-function scatterer we find that the dissipative coupling eliminates the localization in energy space found previously and leads to a well defined resistive steady state. The scattering introduces reproducible fluctuations around the average Ohmic behavior which are caused by coherent backscattering. Their magnitude depends on the strength of the scattering potential and decays slowly for large voltages. The associated correlation energy is determined by the uncertainty of the eigenstates due to the dissipative bath coupling. Thermal averaging leads to a decrease of the conductance fluctuations proportional to T^?1.     

On the theory of quantum interference between inelastic and elastic electron scattering events

The mechanism of weak localization of relatively fast electrons scattered with a fixed energy loss from disordered media is examined. The main focus of this paper is to put forward an explanation why coherent enhancement of electron scattering in the inelastic-scattering channel takes place at angles which differ from π. A simplified kinematic model is proposed to determine the basic properties of the weak localization of electrons in the inelastic scattering channel. The model reproduces easily the range of scattering angles typical of the weak localization of electrons with a fixed energy loss. The procedure does not require calculation of the contribution from the crossed diagrams. The results agree with those based on the dynamical theory associated with the calculation of the crossed and ladder diagrams. It is possible to follow the transition from the new type of weak localization to the ordinary weak localization with decreasing energy loss. The new-type weak localization is in agreement with the regular weak localization if the energy loss is approximately equal to the energy of an optical phonon. Zh. éksp. Teor. Fiz. 111, 1001–1015 (March 1997) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Spin-wave contributions to nuclear magnetic relaxation in magnetic metals

The longitudinal and transverse nuclear magnetic relaxation rates 1/T ₁(T) and 1/T ₂(T) are calculated for three- and two-dimensional (3D and 2D) metallic ferro- and antiferromagnets (FM and AFM) with localized magnetic moments in the spin-wave temperature region. The contribution of the one-magnon decay processes is strongly enhanced in comparison with the standard T-linear Korringa term, especially for the FM case. For the 3D AFM case this contribution diverges logarithmically, the divergence being cut at the magnon gap ω due to magnetic anisotropy, and for the 2D AFM case as ω^-1. The electron-magnon scattering processes yield T ²ln(T/ω) and T ²/ω^1/2-terms in 1/T ₁ for the 3D AFM and 2D FM cases, respectively. The two-magnon (“Raman”) contributions are investigated and demonstrated to be large in the 2D FM case. Peculiarities of the isotropic 2D limit (where the correlation length is very large) are analyzed. Received 29 November 1999 and Received in final form 6 June 2000     

随机介质的相干背散射与局域化研究

研究了不同尺度参数和粒子浓度下的ZnO随机介质相干背散射强度的分布规律,采用时域有限差分法分析了不同浓度随机介质的光场能量空间分布,预测了随机激光器阈值的高低。结果表明:同一折射率的介质随着介质尺寸的增大,相干背散射的带宽变窄,局域化参量kl值相应增大,使得局域化程度呈较大幅度减弱趋势;并且随着介质浓度的增加,相干背散射的带宽变宽,局域化程度增强,阈值降低。相干背散射有着光子局域化的先期特征,现在已成为研究光子局域化出现与否的基本判断依据,对研究光子局域化以及随机激光器具有重要意义。     

Perturbation theory results for the diffuse scattering of light from two-dimensional randomly rough metal surfaces

Abstract

Using the unitarity and reciprocity preserving formulation of Brown et al a perturbation treatment, correct to fourth order in the surface profile function, is given for the scattering of electromagnetic waves from a weakly rough, two-dimensional, random metal surface. In this formulation the boundary conditions on the electromagnetic fields are satisfied using the extinction theorem in conjunction with the Rayleigh hypothesis and the vector equivalent of the Kirchhoff integral. The theory is applied to, and results are presented for, several different types of rough surfaces which are characterized by power spectra that are extensions to two-dimensional random surfaces of the power spectrum of some one-dimensional random surfaces recently fabricated by West and O'Donnell. These surfaces, which can be realized experimentally, favor coherent, interferent, multiple scattering of electromagnetic waves via surface plasmon polaritons in intermediate states, and clearly exhibit enhanced backscattering caused by the surface plasmon polariton mechanism. Theoretical results are presented for silver surfaces at optical wavelengths.