A soluble random-matrix model for relaxation in quantum systems

We study the relaxation of a degenerate two-level system interacting with a heat bath, assuming a random-matrix model for the system-bath interaction. For times larger than the duration of a collision and smaller than the Poincaré recurrence time, the survival probability of still finding the system at timet in the same state in which it was prepared att=0 is exactly calculated.

Partially conserved axial-vector current inS-matrix theory

Mandelstam�s argument that PCAC follows from assigning Lorentz quantum numberM=1 to the massless pion is examined in the context of multiparticle dual resonance model. We construct a factorisable dual model for pions which is formulated operatorially on the harmonic oscillator Fock space along the lines of Neveu-Schwarz model. The model has bothm ? andm ? as arbitrary parameters unconstrained by the duality requirement. Adler self-consistency condition is satisfied if and only if the conditionm?2?m?2=1/2 is imposed, in which case the model reduces to the chiral dual pion model of Neveu and Thorn, and Schwarz. The Lorentz quantum number of the pion in the dual model is shown to beM=0.

Finite size scaling in crossover among different random matrix ensembles in microscopic lattice models

Using numerical diagonalization we study the crossover among different random matrix ensembles (Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE) and Gaussian symplectic ensemble (GSE)) realized in two different microscopic models. The specific diagnostic tool used to study the crossovers is the level spacing distribution. The first model is a one-dimensional lattice model of interacting hard-core bosons (or equivalently spin 1/2 objects) and the other a higher dimensional model of non-interacting particles with disorder and spin-orbit coupling. We find that the perturbation causing the crossover among the different ensembles scales to zero with system size as a power law with an exponent that depends on the ensembles between which the crossover takes place. This exponent is independent of microscopic details of the perturbation. We also find that the crossover from the Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE associated with symmetry breaking introduces a subdominant contribution. We also conjecture that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the dimensionality of the system.

Random matrices: Universality of ESDs and the circular law

Given an n x n complex matrix A, let mu(A)(x, y) := 1/n vertical bar{1 <= i <= n, Re lambda(i) <= x, Im lambda(i) <= y}vertical bar be the empirical spectral distribution (ESD) of its eigenvalues lambda(i) is an element of C, i = l, ... , n. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD mu(1/root n An) of a random matrix A(n) = (a(ij))(1 <= i, j <= n), where the random variables a(ij) - E(a(ij)) are i.i.d. copies of a fixed random variable x with unit variance. We prove a universality principle for such ensembles, namely, that the limit distribution in question is independent of the actual choice of x. In particular, in order to compute this distribution, one can assume that x is real or complex Gaussian. As a related result, we show how laws for this ESD follow from laws for the singular value distribution of 1/root n A(n) - zI for complex z. As a corollary, we establish the circular law conjecture (both almost surely and in probability), which asserts that mu(1/root n An) converges to the uniform measure on the unit disc when the a(ij) have zero mean.

Determinantal point processes in the plane from products of random matrices

We show that the density of eigenvalues for three classes of random matrix ensembles is determinantal. First we derive the density of eigenvalues of product of k independent n x n matrices with i.i.d. complex Gaussian entries with a few of matrices being inverted. In second example we calculate the same for (compatible) product of rectangular matrices with i.i.d. Gaussian entries and in last example we calculate for product of independent truncated unitary random matrices. We derive exact expressions for limiting expected empirical spectral distributions of above mentioned ensembles.

Influence of spatial variability of permeability property on steady state seepage flow and slope stability analysis

In recent years, spatial variability modeling of soil parameters using random field theory has gained distinct importance in geotechnical analysis. In the present Study, commercially available finite difference numerical code FLAC 5.0 is used for modeling the permeability parameter as spatially correlated log-normally distributed random variable and its influence on the steady state seepage flow and on the slope stability analysis are studied. Considering the case of a 5.0 m high cohesive-frictional soil slope of 30 degrees, a range of coefficients of variation (CoV%) from 60 to 90% in the permeability Values, and taking different values of correlation distance in the range of 0.5-15 m, parametric studies, using Monte Carlo simulations, are performed to study the following three aspects, i.e., (i) effect ostochastic soil permeability on the statistics of seepage flow in comparison to the analytic (Dupuit's) solution available for the uniformly constant permeability property; (ii) strain and deformation pattern, and (iii) stability of the given slope assessed in terms of factor of safety (FS). The results obtained in this study are useful to understand the role of permeability variations in slope stability analysis under different slope conditions and material properties. (C) 2009 Elsevier B.V. All rights reserved.

Response of vertically loaded pile in clay: a probabilistic study

This study presents the response of a vertically loaded pile in undrained clay considering spatially distributed undrained shear strength. The probabilistic study is performed considering undrained shear strength as random variable and the analysis is conducted using random field theory. The inherent soil variability is considered as source of variability and the field is modeled as two dimensional non-Gaussian homogeneous random field. Random field is simulated using Cholesky decomposition technique within the finite difference program and Monte Carlo simulation approach is considered for the probabilistic analysis. The influence of variance and spatial correlation of undrained shear strength on the ultimate capacity as summation of ultimate skin friction and end bearing resistance of pile are examined. It is observed that the coefficient of variation and spatial correlation distance are the most important parameters that affect the pile ultimate capacity.

The Ginibre Ensemble and Gaussian Analytic Functions

We show that as n changes, the characteristic polynomial of the n x n random matrix with i.i.d. complex Gaussian entries can be described recursively through a process analogous to Polya's urn scheme. As a result, we get a random analytic function in the limit, which is given by a mixture of Gaussian analytic functions. This suggests another reason why the zeros of Gaussian analytic functions and the Ginibre ensemble exhibit similar local repulsion, but different global behavior. Our approach gives new explicit formulas for the limiting analytic function.

Universal power law in crossover from integrability to quantum chaos

We study models of interacting fermions in one dimension to investigate the crossover from integrability to nonintegrability, i.e., quantum chaos, as a function of system size. Using exact diagonalization of finite-sized systems, we study this crossover by obtaining the energy level statistics and Drude weight associated with transport. Our results reinforce the idea that for system size L -> infinity nonintegrability sets in for an arbitrarily small integrability-breaking perturbation. The crossover value of the perturbation scales as a power law similar to L-eta when the integrable system is gapless. The exponent eta approximate to 3 appears to be robust to microscopic details and the precise form of the perturbation. We conjecture that the exponent in the power law is characteristic of the random matrix ensemble describing the nonintegrable system. For systems with a gap, the crossover scaling appears to be faster than a power law.

Information theory and one-dimensional disordered conductors

A novel method is proposed to treat the problem of the random resistance of a strictly one-dimensional conductor with static disorder. It is suggested, for the probability distribution of the transfer matrix of the conductor, the distribution of maximum information-entropy, constrained by the following physical requirements: 1) flux conservation, 2) time-reversal invariance and 3) scaling, with the length of the conductor, of the two lowest cumulants of ζ, where = sh2ζ. The preliminary results discussed in the text are in qualitative agreement with those obtained by sophisticated microscopic theories.

Information theory and resistance fluctuations in one-dimensional disordered conductors

A novel method is proposed to treat the problem of the random resistance of a strictly one-dimensional conductor with static disorder. For the probability distribution of the transfer matrix R of the conductor we propose a distribution of maximum information entropy, constrained by the following physical requirements: (1) flux conservation, (2) time-reversal invariance, and (3) scaling with the length of the conductor of the two lowest cumulants of ω, where R=exp(iω→⋅Jbhat). The preliminary results discussed in the text are in qualitative agreement with those obtained by sophisticated microscopic theories.

A note on the effective medium theory of random resistive-reactive medium

The effective medium theory for a system with randomly distributed point conductivity and polarisability is reformulated, with attention to cross-terms involving the two disorder parameters. The treatment reveals a certain inconsistency of the conventional theory owing to the neglect of the Maxwell-Wagner effect. The results are significant for the critical resistivity and dielectric anomalies of a binary liquid mixture at the phase separation point.

Optimal maintenance policy for equipment subject to random deterioration and random failure A modern control theory approach.

The stochastic version of Pontryagin's maximum principle is applied to determine an optimal maintenance policy of equipment subject to random deterioration. The deterioration of the equipment with age is modelled as a random process. Next the model is generalized to include random catastrophic failure of the equipment. The optimal maintenance policy is derived for two special probability distributions of time to failure of the equipment, namely, exponential and Weibull distributions Both the salvage value and deterioration rate of the equipment are treated as state variables and the maintenance as a control variable. The result is illustrated by an example

Hanle-Zeeman redistribution matrix. I. Classical theory expressions in the laboratory frame

Polarized scattering in spectral lines is governed by a 4; 4 matrix that describes how the Stokes vector is scattered and redistributed in frequency and direction. Here we develop the theory for this redistribution matrix in the presence of magnetic fields of arbitrary strength and direction. This general magnetic field case is called the Hanle- Zeeman regime, since it covers both of the partially overlapping weak- and strong- field regimes in which the Hanle and Zeeman effects dominate the scattering polarization. In this general regime, the angle-frequency correlations that describe the so-called partial frequency redistribution (PRD) are intimately coupled to the polarization properties. We develop the theory for the PRD redistribution matrix in this general case and explore its detailed mathematical properties and symmetries for the case of a J = 0 -> 1 -> 0 scattering transition, which can be treated in terms of time-dependent classical oscillator theory. It is shown how the redistribution matrix can be expressed as a linear superposition of coherent and noncoherent parts, each of which contain the magnetic redistribution functions that resemble the well- known Hummer- type functions. We also show how the classical theory can be extended to treat atomic and molecular scattering transitions for any combinations of quantum numbers.

Modified density matrix renormalization group algorithm for the zigzag spin-1/2 chain with frustrated antiferromagnetic exchange: Comparison with field theory at large J(2)/J(1)

A modified density matrix renormalization group (DMRG) algorithm is applied to the zigzag spin-1/2 chain with frustrated antiferromagnetic exchange J(1) and J(2) between first and second neighbors. The modified algorithm yields accurate results up to J(2)/J(1) approximate to 4 for the magnetic gap Delta to the lowest triplet state, the amplitude B of the bond order wave phase, the wavelength lambda of the spiral phase, and the spin correlation length xi. The J(2)/J(1) dependences of Delta, B, lambda, and xi provide multiple comparisons to field theories of the zigzag chain. The twist angle of the spiral phase and the spin structure factor yield additional comparisons between DMRG and field theory. Attention is given to the numerical accuracy required to obtain exponentially small gaps or exponentially long correlations near a quantum phase transition.