Random walks in one-dimensional environments with feedback-coupling

Random walks in one-dimensional environments with an additional dynamical feedback-coupling is analyzed numerically. The feedback introduced via a generalized master equation is controlled by a memory kernel of strength the explicit form of which is motivated by arguments used in mode-coupling theories. Introducing several realizations of the feedback mechanism within the simulations we obtain for a negative memory term, , superdiffusion in the long time limit while a positive memory leads to localization of the particle. The numerical simulations are in agreement with recent predictions based on renormalization group techniques. A slight modification of the model including an exponentially decaying memory term and some possible applications for glasses and supercooled liquids are suggested. The relation to the true self-avoiding is discussed. Received 16 September 1999 and Received in final form 27 December 1999     

Random walks on lattices with randomly distributed traps I. The average number of steps until trapping

For a random walk on a lattice with a random distribution of traps we derive an asymptotic expansion valid for smallq for the average number of steps until trapping, whereq is the probability that a lattice point is a trap. We study the case of perfect traps (where the walk comes to an end) and the extension obtained by letting the traps be imperfect (i.e., by giving the walker a finite probability to remain free when stepping on a trap). Several classes of random walks of varying dimensionality are considered and special care is taken to show that the expansion derived is exact up to and including the last term calculated. The numerical accuracy of the expansion is discussed.     

Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries

The general study of random walks on a lattice is developed further with emphasis on continuous-time walks with an asymmetric bias. Continuous time walks are characterized by random pauses between jumps, with a common pausing time distribution(t). An analytic solution in the form of an inverse Laplace transform for P(l, t), the probability of a walker being atl at timet if it started atl _o att=0, is obtained in the presence of completely absorbing boundaries. Numerical results for P(l, t) are presented for characteristically different (t), including one which leads to a non-Gaussian behavior for P(l, t) even for larget. Asymptotic results are obtained for the number of surviving walkers and the mean l showing the effect of the absorption at the boundary.This study was partially supported by ARPA and monitored by ONR(N00014-17-0308).     

Random walks on lattices with points of two colors. II. Some rigorous inequalities for symmetric random walks

We continue our investigation of a model of random walks on lattices with two kinds of points, black and white. The colors of the points are stochastic variables with a translation-invariant, but otherwise arbitrary, joint probability distribution. The steps of the random walk are independent of the colors. We are interested in the stochastic properties of the sequence of consecutive colors encountered by the walker. In this paper we first summarize and extend our earlier general results. Then, under the restriction that the random walk be symmetric, we derive a set of rigorous inequalities for the average length of the subwalk from the starting point to a first black point and of the subwalks between black points visited in succession. A remarkable difference in behavior is found between subwalks following an odd-numbered and subwalks following an evennumbered visit to a black point. The results can be applied to a trapping problem by identifying the black points with imperfect traps.     

First-order transitions in one-dimensional systems with local couplings

We point out the existence of first-order phase transitions in a family of one-dimensional classical spin systems. The relevant features of such models are that they involve only local (but complex) interactions and that the corresponding transfer matrices are self-adjoint operators. Moreover, for a wide range of coupling parameters the models satisfy the reflection positivity condition. The generalization for continuous spin systems enjoys similar properties.     

Random walks with ‘spontaneous emission’ on lattices with periodically distributed imperfect traps

We study random walks on d-dimensional lattices with periodically distributed traps in which the walker has a finite probability per step of disappearing from the lattice and a finite probability of escaping from a trap. General expressions are derived for the total probability that the walk ends in a trap and for the moments of the number of steps made before this happens if it does happen. The analysis is extended to lattices with more types of traps and to a model where the trapping occurs during special steps. Finally, the Green's function at the origin G(0; z) for a finite lattice with periodic boundary conditions, which enters into the main expressions, is studied more closely. A generalization of an expression for G(0; 1) for the square lattice given by Montroll to values of z different from, but close to, 1 is derived.     

Random walks on periodic and random lattices. II. Random walk properties via generating function techniques

We investigate the random walk properties of a class of two-dimensional lattices with two different types of columns and discuss the dependence of the properties on the densities and detailed arrangements of the columns. We show that the row and column components of the mean square displacement are asymptotically independent of the details of the arrangement of columns. We reach the same conclusion for some other random walk properties (return to the origin and number of distinct sites visited) for various periodic arrangements of a given relative density of the two types of columns. We also derive exact asymptotic results for the occupation probabilities of the two types of distinct sites on our lattices which validate the basic conjecture on bond and step ratios made in the preceding paper in this series.Supported in part by a grant from Charles and Renée Taubman and by the National Science Foundation, Grant CHE 78-21460.     

Quantum walks with an anisotropic coin II: scattering theory

We perform the scattering analysis of the evolution operator of quantum walks with an anisotropic coin, and we prove a weak limit theorem for their asymptotic velocity. The quantum walks that we consider include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. Our analysis is based on an abstract framework for the scattering theory of unitary operators in a two-Hilbert spaces setting, which is of independent interest.

     

Controlled excitation of resonance self-oscillations in one-dimensional distributed systems

On the basis of the method of equivalent linearization combined with the method of moments, laws of self-oscillation excitation are obtained that provide the modes with maximum intensity of resonance (or quasi-resonance) oscillations in one-dimensional systems with distributed parameters. A restriction of a general type is imposed on the law of excitation. In the particular case of an integral quadratic restriction, the law of excitation leads to the generation of purely harmonic self-oscillations. The use of an extended (multiplicatively stabilizing) control provides the uniqueness and stability of the quasi-optimal mode of self-oscillation.     

Analytic theory of electric field depinning in the one-dimensional Fukuyama-Lee-Rice hamiltonian

Using Feigel'man's method in the theory of one-dimensional random systems we have evaluated analytically the depinning electric fieldE _T and the static dielectric constant ₀ for the Fukuyama-Lee-Rice hamiltonian in the weak pinning limit and for low temperatures. This is accomplished by solving a Fokker-Planck equation for finitedc electric fields in order to determine the field dependent pinning energy. The dielectric constant is found to remain independent of the electric field up to the threshold. The product ₀ E _T is also evaluated and compared with other theories.     

Absence of phase transitions in certain one-dimensional long-range random systems

An Ising chain is considered with a potential of the formJ(i, j)/|i–j|, where theJ(i, j) are independent random variables with mean zero. The chain contains both randomness and frustration, and serves to model a spin glass. A simple argument is provided to show that the system does not exhibit a phase transition at a positive temperature if>1. This is to be contrasted with a ferromagnetic interaction which requires>2. The basic idea is to prove that the surfacefree energy between two half-lines is finite, although the surface energy may be unbounded. Ford-dimensional systems, it is shown that the free energy does not depend on the specific boundary conditions if>(1/2)d.     

A statistical theory for powder EPR in distributed systems

A statistical interpretation is presented for “g strain,” the dominant broadening in the EPR spectra of metallo-proteins. The direct cause of g strain is described by a three-dimensional tensor p, whose principal elements are random variables. The p and g tensors are not necessarily colinear. The observed EPR linewidth results from a distribution in the effective g value as a function of (a) the joint distribution function of the elements of the p tensor and (b) the spatial relationship between the two principal axis systems involved. The theory is reformulated in terms of matrices that facilitate a direct comparison with earlier work. Two previous theories of g strain represent different subsets of the general theory, namely, the case of zero rotation between axis systems and the case with nonzero rotation and full correlation between elements of the p tensor.     

Random walks on lattices with points of two colours. I

This paper is concerned with random walks on lattices with two kinds of points, black and white. The colours of the points are random variables with a translation invariant, but otherwise arbitrary, joint probability distribution. The steps of the walk are independent of the colours. We study the stochastic properties of the length of the subwalk from the starting point to a first black point and of subwalks between points visited in succession, and establish a number of exact relations. These relations can be applied to a trapping problem by identifying the black points with imperfect traps. An example is discussed.     

Random walks with imperfect trapping in the decoupled-ring approximation

We investigate random walks on a lattice with imperfect traps. In one dimension, we perturbatively compute the survival probability by reducing the problem to a particle diffusing on a closed ring containing just one single trap. Numerical simulations reveal this solution, which is exact in the limit of perfect traps, to be remarkably robust with respect to a significant lowering of the trapping probability. We demonstrate that for randomly distributed traps, the long-time asymptotics of our result recovers the known stretched exponential decay. We also study an anisotropic three-dimensional version of our model. We discuss possible applications of some of our findings to the decay of excitons in semiconducting organic polymer materials, and emphasize the crucial influence of the spatial trap distribution on the kinetics. Received 23 July 2001 / Received in final form 14 May 2002 Published online 13 August 2002     

The bond problem in a one-dimensional percolation theory for finite systems

The results of investigations of main characteristics of a one-dimensional percolation theory (percolation threshold, critical exponents of correlation radius and specific heat, and free energy) are presented for the bond and site problems. For the first time it is shown that for a finite-size system the stability condition is fulfilled while the scaling hypothesis is inacceptable for one-dimensional bond problem.     

Dielectric theory of metal-insulator transitions in doped systems

Model dielectric functions for doped systems show how the dopant excitations change from single-particle-like to collective in character as the concentration is increased from a dilute insulating limit to a more concentrated metallic regime.     

Survival probabilities for random walks on lattices with randomly distributed traps

I present here a numerical procedure to compute survival probabilities for random walks on lattices with randomly distributed traps. The procedure has some advantages over existing methods, and its performance is evaluated for the 1D simple random walk, for which some exact results are known. Thereafter, I apply the procedure to 1D random walks with variable step length and to 3D simple random walks.     

Random matrix theory for pseudo-Hermitian systems: Cyclic blocks

We discuss the relevance of random matrix theory for pseudo-Hermitian systems, and, for Hamiltonians that break parity P and time-reversal invariance T. In an attempt to understand the random Ising model, we present the treatment of cyclic asymmetric matrices with blocks and show that the nearest-neighbour spacing distributions have the same form as obtained for the matrices with scalar entries. We also summarize the theory for random cyclic matrices with scalar entries. We have also found that for block matrices made of Hermitian and pseudo-Hermitian sub-blocks of the form appearing in Ising model depart from the known results for scalar entries. However, there is still similarity in trends even in log-log plots.     

Quantum field theory of phase transitions in Fermi systems

The theory of fermion phase transitions is reviewed from a unified field theoretic standpoint, based on the diagrammatic perturbation expansion of a generalized matrix propagator. Transitions from a normal to a condensed phase are characterized by the spontaneous appearance of long-range order and (in the presence of a suitable infinitesimal external field) broken symmetry. This is illustrated by the ferromagnetic, solid, superconducting and spindensity wave ground states. The phenomenon is explained qualitatively as caused by the creation of a long-range internal field, F, due to the interactions between particles. This field establishes long-range order in the system, and is in turn itself established by the long-range order, in a self-consistent fashion. The mechanism here is expressed quantitatively in terms of a self-consistent Dyson equation relating a generalized matrix propagator, G, to a proper self-energy matrix, Σ. The off-diagonal elements of G describe ‘anomalous’ propagation processes which are characteristic for the condensed phase, and they yield directly the long-range order parameters. The Σ-matrix is just the potential of the internal field. The method is illustrated by applying it to the ferromagnetic phase of a system with δ-function interaction between particles. Finally, the technique is used to derive the vertex part equation for the transition point.