Lognormal Conductivity Research Articles

Bimodal conductance distribution of Kitaev edge modes in topological superconductors

A two-dimensional superconductor with spin-triplet p-wave pairing supports chiral or helical Majorana edge modes with a quantized (length L-independent) thermal conductance. Sufficiently strong anisotropy removes both chirality and helicity, doubling the conductance in the clean system and imposing a super-Ohmic decay in the presence of disorder. We explain the absence of localization in the framework of the Kitaev Hamiltonian, contrasting the edge modes of the two-dimensional system with the one-dimensional Kitaev chain. While the disordered Kitaev chain has a log-normal conductance distribution peaked at an exponentially small value, the Kitaev edge has a bimodal distribution with a second peak near the conductance quantum. Shot noise provides an alternative, purely electrical method of detection of these charge-neutral edge modes.

Resistivity logging in a multi-scale medium with lognormal conductivity

We derived equations of effective conductivity coefficients to estimate mean current density and its variance in resistivity logging problems for isotropic multi-scale porous media with lognormal conductivity distribution. The equations have quite a simple form if the medium parameters satisfy refined Kolmogorov’s similarity hypothesis. The results of the refined perturbation theory obtained using subgrid modeling are compared with the results of numerical simulation and the conventional perturbation theory.

Modeling flow and transport in highly heterogeneous three‐dimensional aquifers: Ergodicity, Gaussianity, and anomalous behavior—1. Conceptual issues and numerical simulations

Uniform flow of mean velocity U takes place in a highly heterogeneous, isotropic, aquifer of lognormal conductivity distribution (variance σY2, integral scale I). A conservative solute is injected instantaneously over an area A0 at x = 0, normal to the mean flow, in a flux‐proportional mode. Longitudinal spreading is caused by advection by the fluid velocity and is quantified with the aid of the mass flux μ(t, x) through fixed control planes at x. An equivalent macrodispersivity is defined in terms of the traveltime variance. The flow and transport are solved numerically in three realizations of the conductivity field with σY2 = 2, 4, 8, respectively. The medium is modeled by a collection of a large number N = 100,000 of spherical inclusions whose conductivities are drawn at random. Transport is simulated by tracking 40,000 particles originating at a large injection area (A0 ≃ 2000 I2) and for travel distance x ≤ 121 I. It is found that the mass flux has a highly skewed time distribution because of the late arrival of solute particles that are moving through low‐conductivity blocks. The tail leads to large values of the equivalent macrodispersivity, which is highly dependent on cutoffs corresponding to the arrival of even 0.999 or 0.995 of the total mass. Furthermore, the tail is nonergodic, as it depends on the plume size. Transport appears to be anomalous in the considered interval, although by the central limit theorem it has to tend asymptotically to Fickianity and Gaussianity.

Finite volume multiscale finite element method for solving the groundwater flow problems in heterogeneous porous media

In this paper we present a numerical method, the finite volume multiscale finite element method (FVMSFEM), for solving the groundwater flow problems in heterogeneous porous media spanning over many scales. This method is based on an efficient coupling between the finite volume discretization and the multiscale finite element base functions. It can efficiently capture the large‐scale structure of the solution on a coarse grid without resolving all the small‐scale features and is locally conservative. The underlying idea is to estimate the macroscopic fluxes across the finite control volume interface segments, which bring the local small‐scale information of the medium property to the large scales, by employing the multiscale finite element base functions. We describe the strategy for constructing such a method on the basis of the transient flow problems in porous media. Numerical experiments are carried out for groundwater flow in porous media with a random lognormal conductivity field to demonstrate the efficiency and accuracy of the developed method.

The Velocity Variance Tensor in a Plane Seepage Flow

In the second-order perturbation theory approximation, the longitudinal and transverse components of the velocity variance tensor are calculated for a plane flow through a randomly heterogeneous porous medium with a log-normal conductivity distribution.

Steady two‐dimensional groundwater flow through many elliptical inhomogeneities

A new analytic element solution has been derived for steady two‐dimensional groundwater flow through an aquifer that contains an arbitrary number of elliptical inhomogeneities. The hydraulic conductivity of each inhomogeneity is homogeneous and differs from the conductivity of the homogeneous background. In addition to elliptical inhomogeneities, other elements (such as wells and line sinks) may be present. The method is based on a separable form of the solution for Laplace's differential equation in elliptical coordinates. The piezometric head and the stream function, expressed as continuous spatial functions (as components of the complex potential), may be obtained up to machine accuracy regardless of the shape, size, orientation, and conductivity of the elliptical inhomogeneities. Components of the discharge vector are expressed in a similar manner, using a complex discharge function. Problems with 10,000 or more inhomogeneities can be solved using parallel computing on distributed memory supercomputer clusters. Two examples are included to demonstrate the precision and capabilities of the method. The second example is used to perform a preliminary study of contaminant transport in a highly heterogeneous formation of lognormal conductivity distribution. The results of the transport study are compared with recent theoretical and numerical results that are based on circular inhomogeneities. The new results with elliptical inhomogeneities confirm the findings based on circular inhomogeneities, including a long dispersion setting time and a zero value of the asymptotic transverse dispersivity.

Flow and transport in highly heterogeneous formations: 3. Numerical simulations and comparison with theoretical results

In parts 1 [Dagan et al., 2003] and 2 [Fiori et al., 2003] a multi‐indicator model of heterogeneous formations is devised in order to solve flow and transport in highly heterogeneous formations. The isotropic medium is made up from circular (2‐D) or spherical (3‐D) inclusions of different conductivities K, submerged in a matrix of effective conductivity. This structure is different from the multi‐Gaussian one, even for equal log conductivity distribution and integral scale. A snapshot of a two‐dimensional plume in a highly heterogeneous medium of lognormal conductivity distribution shows that the model leads to a complex transport picture. The present study was limited, however, to investigating the statistical moments of ergodic plumes. Two approximate semianalytical solutions, based on a self‐consistent model (SC) and on a first‐order perturbation in the log conductivity variance (FO), are used in parts 1 and 2 in order to compute the statistical moments of flow and transport variables for a lognormal conductivity pdf. In this paper an efficient and accurate numerical procedure, based on the analytic‐element method [Strack, 1989], is used in order to validate the approximate results. The solution satisfies exactly the continuity equation and at high‐accuracy the continuity of heads at inclusion boundaries. The dimensionless dependent variables depend on two parameters: the volume fraction n of inclusions in the medium and the log conductivity variance σY2. For inclusions of uniform radius, the largest n was 0.9 (2‐D) and 0.7 (3‐D), whereas the largest σY2 was equal to 10. The SC approximation underestimates the longitudinal Eulerian velocity variance for increasing n and increasing σY2 in 2‐D and, to a lesser extent, in 3‐D, as compared to numerical results. The FO approximation overestimates these variances, and these effects are larger in the transverse direction. The longitudinal velocity pdf is highly skewed and negative velocities are present at high σY2, especially in 2‐D. The main results are in the comparison of the macrodispersivities, computed with the aid of the Lagrangian velocity covariances, as functions of travel time. For the longitudinal macrodispersivity, the SC approximation yields results close to the numerical ones in 2‐D for n = 0.4 but underestimates them for n = 0.9. The asymptotic, large travel time values of macrodispersivities in the SC and FO approximations are close for low to moderate σY2, as shown and explained in part 1. However, while the slow tendency to Fickian behavior is well reproduced by SC, it is much quicker in the FO approximation. In 3‐D the SC approximation is closer to numerical one for the highest n = 0.7 and the different σY2 = 2, 4, 8, and the comparison improves if molecular diffusion is taken into account. Transverse macrodispersivity for small travel times is underestimated by SC in 2‐D and is closer to numerical results in 3‐D, whereas FO overestimates them. Transverse macrodispersivity asymptotically tends to zero in 2‐D for large travel times. In 3‐D the numerical simulations lead to a small but persistent transverse macrodispersivity for large travel times, whereas it tends to zero in the approximate solutions. The results suggest that the self‐consistent semianalytical approximation provides a valuable tool to model transport in highly heterogeneous isotropic formations of a 3‐D structure in terms of trajectories statistical moments. It captures effects like slow transition to Fickian behavior and to Gaussian trajectory distribution, which are neglected by the first‐order approximation.

Numerical simulations of preasymptotic transport in heterogeneous porous media: Departures from the Gaussian limit

The objective of this work is to determine whether conventional solute macrodispersion theories adequately predict the behavior of individual subsurface plumes over time and length scales relevant to practical risk assessment and remediation activities. This issue is studied with a set of high‐resolution numerical simulations of conservative tracers moving through saturated two‐dimensional heterogeneous conductivity fields. The simulation experiments are designed to mimic long‐term field studies. Spatially correlated statistically stationary lognormal conductivity realizations are generated with a Fourier transform procedure. Steady state velocity solutions are calculated for these fields using an accurate Darcian solver. Velocity contour plots reveal the presence of disconnected networks of preferential pathways over a range of correlation lengths. Reverse flow cells are rare. The velocity probability density functions have exponential tails and strong longitudinal asymmetries. Solute concentrations are derived from the simulated velocity fields with an accurate taut‐spline transport code that minimizes numerical dispersion. The resulting plumes are tracked for travel distances of over one hundred spatial correlation lengths, corresponding to scales of practical interest. First moments and macrodispersivities, which measure the location and extent of the plume, are reasonably well approximated by conventional Fickian theories but continue to vary after long travel distances. This temporal variability highlights the slow convergence of the plume moments to Gaussian limits. Calculations of the relative dilution index, which is a measure of the plume mixing state, indicate strongly non‐Gaussian behavior. Other measures of plume structure suggested by anomalous dispersion theories also reveal the persistence of non‐Gaussian behavior after long travel distances. These measures appear to be more sensitive to non‐Gaussian behavior than the spatial moments. Taken together, the simulation results suggest that conservative solute plumes moving through statistically stationary random media may not converge to Gaussian limits even after traveling hundreds of log conductivity correlation scales.

Effective Conductivity of an Isotropic Heterogeneous Medium of Lognormal Conductivity Distribution

The study aims at deriving the effective conductivity Kef of a three-dimensional heterogeneous medium whose local conductivity K(x) is a stationary and isotropic random space function of lognormal distribution and finite integral scale IY. We adopt a model of spherical inclusions of different K, of lognormal pdf, that we coin as a multi-indicator structure. The inclusions are inserted at random in an unbounded matrix of conductivity K 0 within a sphere $\Omega $, of radius R 0, and they occupy a volume fraction n. Uniform flow of flux $% U_{\infty }$ prevails at infinity. The effective conductivity is defined as the equivalent one of the sphere $\Omega ,$ under the limits $n\rightarrow 1$ and $R_{0}/I_{Y}\rightarrow \infty .$ Following a qualitative argument, we derive an exact expression of Kef by computing it at the dilute limit $% n\rightarrow 0.$ It turns out that Kef is given by the well-known self-consistent or effective medium argument. The above result is validated by accurate numerical simulations for $\sigma _{Y}^{2}$ $\leq $ 10 and for spheres of uniform radii. By using a faced-centered cubic lattice arrangement, the values of the volume fraction are in the interval 0<n < 0.7. The simulations are carried out by the means of an analytic element procedure. To exchange space and ensemble averages, a large number N=10000 of inclusions is used for most simulations. We surmise that the self-consistent model is an exact one for this type of medium that is different from the multi-Gaussian one.

EXACTLY SOLVABLE SCALING THEORY OF CONDUCTION IN DISORDERED WIRES

Recent developments in the scaling theory of phase-coherent conduction through a disordered wire are reviewed. The Dorokhov–Mello–Pereyra–Kumar equation for the distribution of transmission eigenvalues has been solved exactly, in the absence of time-reversal symmetry. Comparison with the previous prediction of random-matrix theory shows that this prediction was highly accurate but not exact: the repulsion of the smallest eigenvalues was overestimated by a factor of two. This factor of two resolves several disturbing discrepancies between random-matrix theory and microscopic calculations, notably in the magnitude of the universal conductance fluctuations in the metallic regime, and in the width of the log-normal conductance distribution in the insulating regime.

Higher order correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity distribution : Dagan, G Trans Porous MediaV12, N3, Sept 1993, P279–290

Higher order correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity distribution : Dagan, G Trans Porous MediaV12, N3, Sept 1993, P279–290

Exact solution for the distribution of transmission eigenvalues in a disordered wire and comparison with random-matrix theory.

We consider the complete probability distribution P({${\mathit{T}}_{\mathit{n}}$}) of the transmission eigenvalues ${\mathit{T}}_{1}$,${\mathit{T}}_{2}$,...,${\mathit{T}}_{\mathit{N}}$ of a disordered quasi-one-dimensional conductor (length L much greater than width W and mean free path l). The Fokker-Planck equation which describes the evolution of P with increasing L is mapped onto a Schr\"odinger equation by a Sutherland-type transformation. In the absence of time-reversal symmetry (e.g., because of a magnetic field), the mapping is onto a free-fermion problem, which we solve exactly. The resulting distribution is compared with the predictions of random-matrix theory (RMT) in the metallic regime (L\ensuremath{\ll}Nl) and in the insulating regime (L\ensuremath{\gg}Nl). We find that the logarithmic eigenvalue repulsion of RMT is exact for ${\mathit{T}}_{\mathit{n}}$'s close to unity, but overestimates the repulsion for weakly transmitting channels. The nonlogarithmic repulsion resolves several long-standing discrepancies between RMT and microscopic theory, notably in the magnitude of the universal conductance fluctuations in the metallic regime, and in the width of the log-normal conductance distribution in the insulating regime.

Higher-order correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity distribution

Steady flow of an incompressible fluid takes place in a porous formation of spatially variable hydraulic conductivityK. The latter is regarded as a lognormal stationary random space function and Y=ln(K/K G ), whereK G is the geometric mean ofK, is characterized by its variance σ2 and correlation scale I. Exact results are known for the effective conductivityK eff in one- and two-dimensional flows. In contrast, only a first-order term in a perturbation expansion in σ2 has been derived exactly for the three-dimensional flow. A conjecture has been made in the past onK eff for any σ2, but it was not yet proved exactly. This study derived the exact nonlinear correction, i.e. the termO(σ4) ofK eff, which is found to be the one resulting from the conjecture, strengthening the confidence in it. It is also shown that the self-consistent approximation leads to the exact results for one-dimensional and two-dimensional flows, but underestimates the nonlinear correction ofK eff for in the three-dimensional case.

Conductance statistics in small insulating GaAs:Si wires at low temperature. II: experimental study

We have observed reproducible conductance fluctuations at low temperature in a small GaAs:Si wire driven across the Anderson transition by the application of a gate voltage. We analyse quantitatively the log-normal conductance statistics in terms of truncated quantum fluctuations. Quantum fluctuations due to small changes of the electron energy (controlled by the gate voltage) cannot develop fully due to identified geometrical fluctuations of the resistor network describing the hopping through the sample. The evolution of the fluctuations versus electron energy and magnetic field shows that the fluctuations are non-ergodic, except in the critical insulating region of the Anderson transition, where the localization length is larger than the distance between Si impurities. The mean magnetoconductance is in good accordance with simulations based on the Forward-Directed-Paths analysis, i.e. it saturates to $ {\rm ln} (\sigma (H>1)/\sigma (0))\simeq 1, $ as $ \sigma (0) $ decreases over orders of magnitude in the strongly localized regime.