Bootstrap Multiscale Analysis and Localization¶in Random Media

We introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events. For quantum and classical waves in random media, we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side L with probability higher than 1 − e ^{− L ζ}, for any 0<ζ<1. The starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent, at some sufficiently large scale, with probability bigger than 1 − (d is the dimension). Note that from the same starting hypothesis we get conclusions that are valid for any 0 < ζ < 1. This is achieved by the repeated use of a bootstrap argument. As an application, we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the Hilbert–Schmidt norm, and better estimates on the behavior of the eigenfunctions. Received: 29 November 2000 / Accepted: 21 June 2001     

Semiclassical Estimates¶in Asymptotically Euclidean Scattering

We consider long range semiclassical perturbations of the Laplacian on asymptotically Euclidean manifolds. We obtain precise resolvent estimates under non-trapping assumptions. The novelty lies in a systematic use of geometric microlocal methods.     

Dynamic Localization of a One—Dimensional Quantum Dot Array in an ac Electric Field

We investigate the dynamics of two interaction electrons confined to one-dimensional quantum dot array in an ac electric field.We find that initially localized electrons will remain localized in the absence of coulomb interaction if the ratio of the ac field magnitude to the frequency is a root of the ordinary zero-order Bessel function.In contrast to the case without Coulomb interaction,no matter what the value is,the electrons are delocalized and the delocalization effect depends on the ratio U/ω and eaE/ω,where U is the strength of Coulomb interaction,a is the lattice constant,and E and ω are the ac field amplitude and frequency,respectively.     

One—Dimensional Potentials in q Space

We study one-dimensional potentials in q space and the new features that arise.In particular,we show that the probability of tunnelling of a particle through a barrier or potential step is less than that of the same particle with the same energy in ordinary space which is somehow unexpected.We also show that the tunnelling time for a particle in q space is less than that of the same particle in ordinary space.     

Random Homogenization and Singular Perturbations¶in Perforated Domains

The paper considers the singularly perturbed Dirichlet problem −ɛΔu ^ɛ+u ^ɛ=f in a randomly perforated domain Ω^ɛ, which is obtained from a bounded open set Ω in R ^N after removing many holes of size ɛ ^q . The perforated domain is described in terms of an ergodic dynamical system acting on a probability space. Imposing certain conditions on the domain, the behaviour of u ^ɛ when ɛ→ 0 in Lebesgue spaces L ⁿ (Ω) is studied. Test functions together with the Birkhoff ergodic theorem are the main tools of analysis. The Poisson distribution of holes of size ɛ ^p with the intensity λɛ^{− r} is then considered. The above results apply in some cases; other cases are treated by the Wiener sausage approach. Received: 15 December 1999 / Accepted: 14 April 2000     

Recurrence and Transience of Random Walks¶in Random Environments on a Strip

We explain the necessary and sufficient conditions for recurrent and transient behavior of a random walk in a stationary ergodic random environment on a strip in terms of properties of a top Lyapunov exponent. This Lyapunov exponent is defined for a product of a stationary sequence of positive matrices. In the one-dimensional case this approach allows us to treat wider classes of random walks than before. Received: 15 March 2000 / Accepted: 14 April 2000     

Photon Diffusion in Random Media and Anisotropy of Scattering in the Henyey–Greenstein and Rayleigh–Gans Models

Journal of Experimental and Theoretical Physics - Monte Carlo (MC) numerical simulation is carried out for the intensity of multiply backscattered radiation as a function of the...     

A Spectral Approach to Survival Probabilities in Porous Media

We consider a diffusive process in a bounded domain with heterogeneously distributed traps, reactive regions or relaxing sinks. This is a mathematical model for chemical reactors with heterogeneous spatial distributions of catalytic germs, for biological cells with specific arrangements of organelles, and for mineral porous media with relaxing agents in NMR experiments. We propose a spectral approach for computing survival probabilities which are represented in the form of a spectral decomposition over the Laplace operator eigenfunctions. We illustrate the performances of the approach by considering diffusion inside the unit disk filled with reactive regions of various shapes and reactivities. The role of the spatial arrangement of these regions and its influence on the overall reaction rate are investigated in the long-time regime. When the reactivity is finite, a uniform filling of the disk is shown to provide the highest reaction rate. Although the heterogeneity tends to reduce the reaction rate, reactive regions can still be heterogeneously arranged to get nearly optimal performances.     

Spacing Between Phase Shifts in a Simple¶Scattering Problem

We prove a scattering theoretical version of the Berry–Tabor conjecture: for an almost every surface in a class of cylindrical surfaces of revolution, the large energy limit of the pair correlation measure of the quantum phase shifts is Poisson, that is, it is given by the uniform measure. Received: 15 August 1998 / Accepted: 15 February 1999     

Multiple Scattering Effects of Ag Particles in Random Ag-SiN Film

A layer of 40nm-thick Ag-SiN film with Ag nano-particles embedded and distributed randomly in the SiN thin film were deposited by the method of radiant-frequency magnetron sputtering. Specimens orderly comprising a random Ag-SiN film and an optical phase change recording layer were exposed to a focused laser beam with wavelength of 69Ohm. It is shown that, with a random Ag-SiN layer deposited above the recording layer. Calculation by the finite difference time domain method of a 4Ohm-thick SiN film under a Gaussian beam irradiation has been carried out to simulate the near-field distribution in the film, which showed a huge local near-field intensity enhancement of about 200 times if small Ag particles with diameter of 6 nm were modelled inthe SiN film in the central region of the in cident laser spot.     

Formation of Singularities in One—Dimensional Hydromagnetic Flow

Two results on the formation of singularities in solutions to the system of one-dimensional hydromagnetic dynamics are presented.In particular,it is shown that shocks form from a smooth spatial periodic flow in a finite time if the initial amounts of entropy and the “magnetic field” in each period are smaller than those of sound waves.A quantitative estimate of blow-up time is also given.     

Higher Dimensional Λ-CDM Universe: A Phenomenological Approach with Many Possibilities

In this paper we investigate the Λ-CDM universe by selecting the specific time dependent form of Λ, viz. , in the context of 5-dimensional space time. Time-dependent form of equation of state parameter w is derived along with a possible signature flip of deceleration parameter q. It is also observed that a present age of the Universe, calculated for some specific values of parameter agrees very well with the observational data.     

A Path Integral Approach¶to the Kontsevich Quantization Formula

We give a quantum field theory interpretation of Kontsevich's deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. Its Batalin–Vilkovisky quantization yields a superconformal field theory. The associativity of the star product, and more generally the formality conjecture can then be understood by field theory methods. As an application, we compute the center of the deformed algebra in terms of the center of the Poisson algebra. Received: 10 March 1999 / Accepted: 30 January 2000     

Direct Current Hopping Conductivity in One—Dimensional Nanometre Systems

A one-dimensional random nanocrystalline chain model is established.A dc electron-phonon-field conductance model of electron tunnelling transfer is set up,and a new dc conductance formula in one-dimensional nanometre systems is derived.By calculationg the dc conductivity,the relationship among the electric field,temperature and conductivity is analysed.and the effect of the crystalline grain size and the distrotion of interfacial atoms on the dc conductance is discussed.The result shows that the nanometre system appears the characteristic of negative differential dependence of resistance and temperature at low temperature.The dc conductivity of nanometre systems varies with the change of electric field and trends to rise as the crystalline grain size increases and to decrease as the distroted degree of interfacial atoms increases.     

Random walk in random environment: A counterexample?

We describe a family of random walks in random environments which have exponentially decaying correlations, nearest neighbor transition probabilities which are bounded away from 0, and yet are subdiffusive in any dimensiond<.This author partially supported by NSF grant DMS 83-1080This author partially supported by NSF grant DMS-85-05020 and the Army Research Office through the Mathematical Sciences Institute at Cornell University     

A Coupling Approach¶to Randomly Forced Nonlinear PDE's. I

We develop a coupling approach to prove that a randomly forced dissipative PDE has a unique stationary measure and to study ergodic properties of this measure. Received: 12 March 2001 / Accepted: 27 March 2001     

Universality at the Edge of the Spectrum¶in Wigner Random Matrices

We prove universality at the edge for rescaled correlation functions of Wigner random matrices in the limit n→+∞. As a corollary, we show that, after proper rescaling, the 1^th, 2^nd, 3^rd, etc. eigenvalues of Wigner random hermitian (resp. real symmetric) matrix weakly converge to the distributions established by Tracy and Widom in G.U.E. (G.O.E.) cases. Received: 15 May 1999 / Accepted: 18 May 1999     

Cube–Root Boundary Fluctuations¶for Droplets in Random Cluster Models

For a family of bond percolation models on ℤ² that includes the Fortuin–Kasteleyn random cluster model, we consider properties of the “droplet” that results, in the percolating regime, from conditioning on the existence of an open dual circuit surrounding the origin and enclosing at least (or exactly) a given large area A. This droplet is a close surrogate for the one obtained by Dobrushin, Kotecky and Shlosman by conditioning the Ising model; it approximates an area-A Wulff shape. The local part of the deviation from the Wulff shape (the “local roughness”) is the inward deviation of the droplet boundary from the boundary of its own convex hull; the remaining part of the deviation, that of the convex hull of the droplet from the Wulff shape, is inherently long-range. We show that the local roughness is described by at most the exponent 1/3 predicted by nonrigorous theory; this same prediction has been made for a wide class of interfaces in two dimensions. Specifically, the average of the local roughness over the droplet surface is shown to be O(l ^1/3(log l)^2/3) in probability, where is the linear scale of the droplet. We also bound the maximum of the local roughness over the droplet surface and bound the long-range part of the deviation from a Wulff shape, and we establish the absense of “bottlenecks”, which are a form of self-approach by the droplet boundary, down to scale log l. Finally, if we condition instead on the event that the total area of all large droplets inside a finite box exceeds A, we show that with probability near 1 for large A, only a single large droplet is present. Received: 20 January 2000 / Accepted: 7 August 2001     

Acoustic Localization in Weakly Compressible Elastic Media Permeated with Air Bubbles

The propagation of longitudinal acoustic waves in weakly compressible elastic media permeated with air bubbles is investigated on the basis of the radial pulsation equation of a single bubble. The multiple scattering of waves in such media is rigorously described by using a self-consistent approach. Theoretical results show that there exists strong acoustic localization in a range of frequency slightly above the bubble resonance frequency, even for a very small volume fraction of bubbles. Further study reveals that the localization is in fact attributed to collection behaviour of bubbles, allowing for an efficient cancellation of propagating waves. This is essentially consistent with the known conclusions recently drawn for bubbly liquid by Kou et al. [2003 Appl. Phys. Left. 83 4247]