Two—phase flow in correlated pore—throat random porous media

We have constructed a porous media model in which there are percolation clusters with varying percolation probability P and correlated site-bonds. Taking into account both the pore and the throat geometry, the viscous fingering (VF) in porous media has been investigated by using the standard over-relaxed Gauss-Seidel scheme. The simulation results show that the VF structure varies with the correlation parameter ε, the viscosity ratio M and the percolation probability P. The smaller the correlation parameter ε, the greater the deviation of the normalized size distribution of the invaded throat N_inv(r) from the truncated Rayleigh distribution. For a larger viscosity ratio M, the VF pattern looks like a diffusion-limited-aggregation structure in percolation clusters. The fractal dimension D increases with the increase of the percolation probability P and the correlation parameter ε. The velocity distribution f(α) of VF in percolation clusters is of a parabola-like curve. The tail of the distribution (large α) is longer for a larger correlation parameter ε. For a smaller ε, the distribution is very sharp. The sweep efficiency E decreases along with the decrease of the correlation parameter ε and the increase of the network size L_nz. E has a minimum as L_nz increases up to the maximum no matter what the values of P, M and ε. The E～ L_nz curve has a frozen zone and an active zone. The geometry and the topology of the porous media have strong effects on the displacement processes and the structure of VF.     

Drift,creep and pinning of a particle in a correlated random potential

The motion of a particle in a correlated random potential under the influence of a driving force is investigated in mean field theory. The correlations of the disorder are characterized by a short distance cutoff and a power law decay with exponent γ at large distances. Depending on temperature and γ, drift with finite mobility, creep or pinning is found. This is in qualitative agreement with results in one dimension. This model is of interest not only in view of the motion of particles or manifolds in random media, it also improves the understanding of glassy non-equilibrium dynamics in mean field models. The results, obtained by numerical integration and analytic investigations of the various scaling regimes in this problem, are compared with previous proposals regarding the long time properties of such systems and with replica calculations.     

Statistics of ideal randomly branched polymers in a semi-space

We investigate the statistical properties of a randomly branched 3-functional N-link polymer chain without excluded volume, whose one point is fixed at the distance d from the impenetrable surface in a 3-dimensional space. Exactly solving the Dyson-type equation for the partition function Z(N, d )= N^-θe^γN in 3D, we find the “surface” critical exponent θ = , as well as the density profiles of 3-functional units and of dead ends. Our approach enables to compute also the pairwise correlation function of a randomly branched polymer in a 3D semi-space.     

Polymers in a weak random potential in dimension four: Rigorous renormalization group analysis

Correlation functions of the Edwards model of polymers at weak coupling are defined and studied at the critical point, in dimension four, by a rigorous renormalization group method which validates, at any order, perturbative renormalization group results on their behaviour at large distances. Remainders are controlled by a new argument which enlarges the use of methods of constructive field theory to models of statistical physics.A large part of this work has also included the collaboration of D. Arnaudon     

Slowly moving matter--wave gap soliton propagation in weak random nonlinear potential

We systematically investigate the motion of slowly moving matter--wave gap solitons in a nonlinear potential, produced by the weak random spatial variation of the atomic scattering length. With the weak randomness, we construct an effective-particle theory to study the motion of gap solitons. Based on the effective-particle theory, the effect of the randomness on gap solitons is obtained, and the motion of gap solitons is finally solved. Moreover, the analytic results for the general behaviours of gap soliton motion, such as the ensemble-average speed and the reflection probability depending on the weak randomness are obtained. We find that with the increase of the random strength the ensemble-average speed of gap solitons decreases slowly where the reduction is proportional to the variance of the weak randomness, and the reflection probability becomes larger. The theoretical results are in good agreement with the numerical simulations based on the Gross--Pitaevskii equation.     

Statistics of waves propagating in a random medium

We present a review of part of the results obtained by the authors for the statistics of coherent radiation propagating in a random medium both in the framework of diagrammatic techniques and random matrix theory. Distribution functions for the total transmission coefficient and the angular transmission coefficient for the diffusive transport and the crossover between the diffusive and ballistic regimes are obtained.The authors acknowledge the financial support of the Israeli Academy of Sciences and of the Schottenstein Center.     

Transport in random correlated fields

A review is given of recent developments in the diffusion properties of particles in the presence of local random fields as well as the conductivity of the analog random resistor network. The effect of long-range ferro- and antiferro-type correlations between the local fields on the diffusion and conductivity properties is considered. A physical realization for such spatial correlations is diffusion on linear polymers in the presence of external uniform bias field. For this case a universal diffusion law was found independent of the fractal dimension of the polymer chain or the Euclidean dimension in which the polymer is embedded. Recent results for diffusion in two dimensions in the presence of a special case of correlated local fields are also reviewed.     

Super-exponential extinction of radiation in a negatively correlated random medium

It was shown in recent work that spatial correlations among obstacles of a random, absorbing medium can lead to slower-than-exponential (sub-exponential) extinction of radiation with propagation distance. Exponential decay, described by the Beer-Lambert law, arises in a special case when the medium contains no correlations. A third possibility, examined here, is that of negative correlations which can lead to faster-than-exponential (super-exponential) extinction. Using a Monte Carlo approach, we confirm that sub-exponential decay occurs when the volume-averaged pair correlation function is greater than zero at the scale of interest and that the Beer-Lambert law is recovered when correlations vanish. We also find that when the volume-averaged pair correlation function is negative, super-exponential extinction with propagation distance occurs. These results are of special interest to the problem of radiative transfer in cloudy atmospheres, where the pair correlation function previously has been shown to be negative and positive at different scales.     

Charged bosons in a random potential

It is shown that the interaction of bosons in a random potential in certain cases provides extended states.     

Bose-Einstein condensate in a random potential

An optical speckle potential is used to investigate the static and dynamic properties of a Bose-Einstein condensate in the presence of disorder. With small levels of disorder, stripes are observed in the expanded density profile and strong damping of dipole and quadrupole oscillations is seen. Uncorrelated frequency shifts of the two modes are measured and are explained using a sum-rules approach and by the numerical solution of the Gross-Pitaevskii equation.     

Percolation in a symmetric random potential

We study percolation in a two-dimensional random Gaussian potential and calculate p_c, the critical exponents υ, β, γ, the fractal dimension of the percolating equipotential lines, and the amplitude ratio C⁺/C⁻ of the perlocation susceptibility χ. These quantities including the amplitude ratio match the known results of the two-dimensional lattice percolation models.     

Localization in one dimensional correlated random potentials

The localization length of one-dimensional disordered systems with statistically correlated random potentials is studied both numerically and analytically. The results indicate that the localization length generally increases when the correlation function is positive. In the presence of anticorrelation effects the localization length may be shorter than in the uncorrelated case.     

Two interacting particles in a random potential

We study the scaling of the localization length of two interacting bosons in a one-dimensional random lattice with the single particle localization length. We consider the short-range interaction assuming that the particles interact when located both on the same site. We discuss several regimes, among them one interesting weak Fock space disorder regime. In this regime we obtain a weak logarithmic scaling law. Numerical benchmark data support the absence of any strong enhancement of the two particle localization length.     

Commensurable charge-density waves in a random potential

We study commensurable charge-density wave systems in the presence of random impurities. The results of the microscopic theory, especially the contributions of second order in the impurity fields, are briefly discussed. The lifetime of the excitations is calculated, as well as the response of a soliton to an external field.