Biased random walk on a multidimensional lattice with absorbing boundaries

This paper deals with continuous-time random walk on a hypercubic lattice with absorbing boundaries along the axial planes through the origin. Possible bias in the transition probabilities along any axis is allowed for. Any dynamical model which is solvable in the case of an infinite lattice is shown to be tractable also in the present case. As an example, the exponential holding time model is solved explicitly. Representative numerical results for the probability of return to the starting point and the probability of absorption are presented for the one-dimensional case. It is found that, unlike the absorption which increases with bias towards the boundary, the return probability is independent of the direction of the bias.     

Exactly solvable Fokker-Planck equation with time-dependent nonlinear drift and diffusion coefficients – the Lie-algebraic approach

In this paper we have analytically solved the Fokker-Planck equation (FPE) associated with a fairly large class of multiplicative stochastic processes with time-varying nonliner drift and diffusion coefficients, which has wide applicability in various areas of physics, e.g. nonlinear optics and chemical reaction dynamics. By exploiting the dynamical symmetry of the FPE, we apply the Lie-algebraic approach to derive the time-dependent analytical closed-form solutions. The derived solutions fall into two different categories, namely (i) one with a moving absorbing boundary, and (ii) one with a fixed absorbing boundary at the origin, depending upon the model parameters. The corresponding escape (or survival) probabilities are also evaluated analytically. We believe that not only our analytically exact results can serve as standard models upon which the discussion of more complicated problems can be based, but they can also be useful as a benchmark to test approximate numerical or analytical procedures.     

Probing the statistics of transport in the Hénon Map

The phase space of an area-preserving map typically contains infinitely many elliptic islands embedded in a chaotic sea. Orbits near the boundary of a chaotic region have been observed to stick for long times, strongly influencing their transport properties. The boundary is composed of invariant “boundary circles.” We briefly report recent results of the distribution of rotation numbers of boundary circles for the Hénon quadratic map and show that the probability of occurrence of small integer entries of their continued fraction expansions is larger than would be expected for a number chosen at random. However, large integer entries occur with probabilities distributed proportionally to the random case. The probability distributions of ratios of fluxes through island chains is reported as well. These island chains are neighbours in the sense of the Meiss-Ott Markov-tree model. Two distinct universality families are found. The distributions of the ratio between the flux and orbital period are also presented. All of these results have implications for models of transport in mixed phase space.     

Open Quantum Random Walks on the Half-Line: The Karlin–McGregor Formula,Path Counting and Foster’s Theorem

In this work we consider open quantum random walks on the non-negative integers. By considering orthogonal matrix polynomials we are able to describe transition probability expressions for classes of walks via a matrix version of the Karlin–McGregor formula. We focus on absorbing boundary conditions and, for simpler classes of examples, we consider path counting and the corresponding combinatorial tools. A non-commutative version of the gambler’s ruin is studied by obtaining the probability of reaching a certain fortune and the mean time to reach a fortune or ruin in terms of generating functions. In the case of the Hadamard coin, a counting technique for boundary restricted paths in a lattice is also presented. We discuss an open quantum version of Foster’s Theorem for the expected return time together with applications.     

Dynamics of a two-level system interacting with a random classical field

The dynamics of a particle interacting with a random classical field in a two-well potential is studied by the functional integration method. The probability of particle localization in either of the wells is studied in detail. Certain field-averaged correlation functions for quantum-mechanical probabilities and the distribution function for the probabilities of final states (which can be considered as random variables in the presence of a random field) are calculated. The calculated correlators are used to discuss the dependence of the final state on the initial state. One of the main results of this work is that, although the off-diagonal elements of the density matrix disappear with time, a particle in the system is localized incompletely (wave-packet reduction does not occur), and the distribution function for the probability of finding particle in one of the wells is a constant at infinite time.     

The two-state random walk

We develop asymptotic results for the two-state random walk, which can be regarded as a generalization of the continuous-time random walk. The two-state random walk is one in which a particle can be in one of two states for random periods of time, each of the states having different spatial transition probabilities. When the sojourn times in each of the states and the second moments of transition probabilities are finite, the state probabilities have an asymptotic Gaussian form. Several known asymptotic results are reproduced, such as the Gaussian form for the probability density of position in continuous-time random walks, the time spent in one of these states, and the diffusion constant of a two-state diffusing particle.     

Competitive trapping effects in a set of partially absorbing traps

This note contains a formalism for calculating properties of random walks in the presence of a set of partially absorbing traps. The properties that are considered are the probability of trapping at a specific point and the survival probability as a function of step number. The results are expressed in terms of determinants, but approximations to these can be found.     

On a coupled random walk process specified by a Hamiltonian

A coupled random walk process specified by an effective Hamiltonian in a potential field is proposed. The Hamiltonian is expressed in terms of a set of jumping probabilities which characterize the random walk processes. The steady state is expressed by the Hamiltonian. Conditions for the Hamiltonian to be reduced to the Ginzburg-Landau type are discussed.     

Inequalities in Entanglement Percolation

We present proofs of two results concerning entanglement in three-dimensional bond percolation. Firstly, the critical probability for entanglement with free boundary conditions is strictly less than the critical probability for connectivity percolation. (The proof presented here is a detailed justification of the ideas sketched in Aizenman and Grimmett.) Secondly, under the hypothesis that the critical probabilities for entanglement with free and wired boundary conditions are different, for p between the two critical probabilities, the size of the entangled cluster at the origin with free boundary conditions does not have exponentially decaying tails.     

Spatial structure and stability based on random walks

A general expression for a recursion formula which describes a random walk with coupled modes is given. In this system, the random walker is specified by the jumping probabilities P⁺ and P^– which depend on the modes. The transition probability between the modes is expressed by a jumping probabilityR ^(ij) (orr _ij). With the aid of this recursion formula, spatial structures of the steady state of a coupled random walk are studied. By introducing a Liapunov function and entropy, it is shown that the stability condition for the present system can be expressed as the principle of the extremum entropy production.On leave of absence from Tohoku University, Department of Applied Science, Faculty of Engineering, Sendai, 980 Japan.     

Potentials for almost Markovian random fields

A positive almost Markovian random field is a probability measure on a lattice gas whose finite set conditional probabilities are continuous and positive. We show that each such random field has a potential and in the translation invariant case an absolutely convergent potential. We give a criterion for determining which random fields correspond to pair potentials, or in generaln-body potentials. We show that two translation invariant positive almost Markovian random fields have the same finite set conditional probabilities if and only if one minimizes the specific free energy of the other.     

Controlling Chaos Probability of a Bose-Einstein Condensate in a Weak Optical Superlattice

The spatial chaos probability of a Bose-Einstein condensate perturbed by a weak optical superlattice is studied. It is demonstrated that the spatial. chaotic solution appears with a certain probability in a given parameter region under a random boundary condition. The effects of the lattice depths and wave vectors on the chaos probability are illustrated, and different regions associated with different chaos probabilities are found. This suggests a feasible scheme for suppressing and strengthening chaos by adjusting the optical superlattice experimentaJly.     

Asymptotic survival probability of random walks on a semi-infinite line

Symmetric and asymmetric random walks on a segment (−∞,T>0) of the real line are considered. There is a non-zero probability for the random walk to get absorbed at a site it visits. We derive for such random walks, expressions for survival probabilities in the asymptotic limit ofT→∞. An application of this asymptotic formulation to the problem of radiation transport through thick shields is presented.     

Activated Random Walkers: Facts, Conjectures and Challenges

We study a particle system with hopping (random walk) dynamics on the integer lattice ? ^d . The particles can exist in two states, active or inactive (sleeping); only the former can hop. The dynamics conserves the number of particles; there is no limit on the number of particles at a given site. Isolated active particles fall asleep at rate λ>0, and then remain asleep until joined by another particle at the same site. The state in which all particles are inactive is absorbing. Whether activity continues at long times depends on the relation between the particle density ζ and the sleeping rate λ. We discuss the general case, and then, for the one-dimensional totally asymmetric case, study the phase transition between an active phase (for sufficiently large particle densities and/or small λ) and an absorbing one. We also present arguments regarding the asymptotic mean hopping velocity in the active phase, the rate of fixation in the absorbing phase, and survival of the infinite system at criticality. Using mean-field theory and Monte Carlo simulation, we locate the phase boundary. The phase transition appears to be continuous in both the symmetric and asymmetric versions of the process, but the critical behavior is very different. The former case is characterized by simple integer or rational values for critical exponents (β=1, for example), and the phase diagram is in accord with the prediction of mean-field theory. We present evidence that the symmetric version belongs to the universality class of conserved stochastic sandpiles, also known as conserved directed percolation. Simulations also reveal an interesting transient phenomenon of damped oscillations in the activity density.     

Boundary-induced nonequilibrium phase transition into an absorbing state

We demonstrate that absorbing phase transitions in one dimension may be induced by the dynamics of a single site. As an example, we consider a one-dimensional model of diffusing particles, where a single site at the boundary evolves according to the dynamics of a contact process. As the rate for offspring production at this site is varied, the model exhibits a phase transition from a fluctuating active phase into an absorbing state. The universal properties of the transition are analyzed by numerical simulations and approximation techniques.     

Phase transition in a four-dimensional random walk with application to medical statistics

A random walk in a piecewise homogeneous medium can exhibit a variety of asymptotic behaviors. In particular, it may lodge strictly in one region or divide in probability among several. This will depend upon the parameters describing (a) the walk, (b) the interregion boundary, and (c) the initial location of the walk. We analyze from this point of view a special four-dimensional walk on an integer lattice with two homogeneous regions separated by a hyperplane of codimension 1. The walk represents a continuing sequence of clinical trials of two drugs of unknown success probabilities and the two regions represent the Bayes-derived criterion as to which drug to try next. The demarcation in the parameter space of success probabilities and initial coordinates between one- and two-region asymptotics is mapped out analytically in several special cases and supporting numerical evidence given in the general case.     

Survival Probability of the Néel State in Clean and Disordered Systems: An Overview

In this work we provide an overview of our recent results about the quench dynamics of one-dimensional many-body quantum systems described by spin-1/2 models. To illustrate those general results, here we employ a particular and experimentally accessible initial state, namely the Néel state. Both cases are considered: clean chains without any disorder and disordered systems with static random on-site magnetic fields. The quantity used for the analysis is the probability for finding the initial state later in time, the so-called survival probability. At short times, the survival probability may decay faster than exponentially, Gaussian behaviors and even the limit established by the energy-time uncertainty relation are displayed. The dynamics at long times slows down significantly and shows a powerlaw behavior. For both scenarios, we provide analytical expressions that agree very well with our numerical results.     

Linear One-Step Processes with Artificial Boundaries

An artificial absorbing boundary is introduced in a linear birth and death stochastic process in order to understand the long time behavior of an ecological community. The solution is obtained by means of a spectral resolution of the probability distribution. A more general linear process with a coefficient of arbitrary strength near the boundary both with absorbing and with reflecting boundary conditions is also studied.     

Exact dynamics of a continuous time random walker in the presence of a boundary: beyond the intuitive boundary condition approach

We derive the exact dynamics of a random walker with arbitrary non-Markovian transport and reaction rate distribution at a boundary, and present exact solutions in the continuum limit. We find that the ultimate escape probability of the particle is independent of the transport mechanism in contradiction to the long-standing belief based on the conventional approach. We also find a phase transition in the relaxation kinetics associated with the heterogeneity of the transport media.     

Local production rate of an “enzyme kinetics” based on coupled random walks

Based on the coupled random walk process, a local production rate of an enzyme kinetics is studied. A substrate as a walker is specified by a set of the jumping probabilities between the sites and between the modes. The modes represent the states of the substrate before, in and after the reaction. The local production rate is explicitly given and its position dependence is studied in terms of jumping probabilities. Qualitatively, the deviations from the Michaelis-Menten's expression are discussed.