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There are many current applications of the continuous-time random walk (CTRW), particularly in describing kinetic and transport processes in different chemical and biophysical phenomena. We derive exact solutions for the Laplace transforms of the propagators for non-Markovian asymmetric one-dimensional CTRW’s in an infinite space and in the presence of an absorbing boundary. The former is used to produce exact results for the Laplace transforms of the first two moments of the displacement of the random walker, the asymptotic behavior of the moments as t→∞, and the effective diffusion constant. We show that in the infinite space, the propagator satisfies a relation that can be interpreted as a generalized fluctuation theorem since it reduces to the conventional fluctuation theorem at large times. Based on the Laplace transform of the propagator in the presence of an absorbing boundary, we derive the Laplace transform of the survival probability of the random walker, which is then used to find the mean lifetime for terminated trajectories of the random walk. [ABSTRACT FROM AUTHOR]

A general formalism for treating lateral diffusion in a multilayer medium is developed. The formalism is based on the relation between the lateral diffusion and the distribution of the cumulative residence time, which the diffusing particle spends in different layers. We exploit this fact to derive general expressions which give the global and local time-dependent diffusion coefficients in terms of the average cumulative times spent by the particle in different layers and the probabilities of finding the particle in different layers, respectively. These expressions are used to generalize two recently obtained results: (a) A solution for the short-time behavior of the lateral diffusion coefficient in two layers separated by a permeable membrane obtained by a perturbation theory is extended to the entire range of time. (b) A solution for the time-dependent diffusion coefficient of a ligand, which repeatedly dissociates and rebinds to sites on a planar surface, obtained under the assumption that the medium above the surface is infinite, is generalized to allow for the medium layer of finite thickness. For the latter problem we derive an expression for the Fourier-Laplace transform of the propagator in terms of the double Laplace transform of the probability density of the cumulative residence time spent by the ligand in the medium layer. [ABSTRACT FROM AUTHOR]

Previously, the authors have calculated a number of properties of diffusion in the presence of partially reacting boundaries, in which the probability that a particle impinging on the boundary will react is not a constant. This is reflected in the appearance of a boundary condition nonlocal in time. Here a number of consequences for kinetic behavior of such a nonlocal boundary condition are derived. The circumstances under which results derived from the Smoluchowski model or its generalization are asymptotically valid are discussed, as well as conditions that lead to deviations from the predictions of such a theory. [ABSTRACT FROM AUTHOR]

The radiation boundary condition was originally proposed by Collins and Kimball as a means of avoiding the unphysical prediction of the Smoluchowski model for reaction rates that the calculated rate in three dimensions k(t) has the property k(0)=∞. A microscopic model that can be used to derive the boundary conditions uses the tacit assumption that an encounter between two molecules A and B gives rise to a reaction with a probability α<1. We consider a non-Markovian model in which the probability that exactly n encounters between A and B are required to produce a reaction is equal to θn. We show that when the expected number of such encounters is finite, one gets the usual radiation boundary condition. When the expected number is infinite, one finds a boundary condition that is nonlocal in time. The extension of our analysis to higher dimensions as well as to the Smoluchowski equation is readily generated. [ABSTRACT FROM AUTHOR]

We consider a reversible dissociation–recombination reaction in solution which is described by a distribution of waiting times rather than a single dissociation rate constant. This is a non-Markovian generalization of the backreaction boundary condition. We formulate the new boundary condition in terms of the residence time in the bound state and illustrate the theory by assuming a stable-law density for the residence time. Explicit expressions are found for the Laplace transform of the survival probability in one and three dimensions, which can be inverted analytically for special values of the stable-law parameter α and numerically for other values of α. We derive the long-time behavior of the survival probability for arbitrary α, and note that the survival probability undergoes a first-order phase transition in one dimension, in which its asymptotic value changes abruptly at α=1/2. In three dimensions it undergoes a second-order phase transition at α=1, in which only the asymptotic slope of the survival probability changes discontinuously. [ABSTRACT FROM AUTHOR]

There are many polymeric materials whose dielectric properties can be derived from the Williams–Watts relaxation function [lowercase_phi_synonym]α(t)=exp [-(t/τ)α] . In this paper we propose a method for estimating the parameters α, τ, and ε(0)-ε(∞), from dielectric loss data. In contrast to earlier techniques the parameters can be estimated one at a time. Simple approximations to the required functions are given accurate to better than 1%. [ABSTRACT FROM AUTHOR]

Most models of absorption sites for random walks or diffusion processes fall into one of two categories: (1) Perfect absorption, in which every encounter of a random walker with a trap produces a trapping event, and (2) imperfect absorption in which an encounter leads to a trapping event with probability α<1. We introduce the notion of a non-Markovian trap characterized by a set of probabilities {f j}, where f j is the probability that the jth encounter leads to a trapping event. Some consequences of this assumption are examined in the context of a one-dimensional trapping problem. It is shown that when the f j have an associated finite first moment the asymptotic survivial probability goes like n1/2 exp(-an1/3) where n is the step number and a is a constant. This is equivalent to the results one would obtain with a Markovian model. However, when f j is asymptotically proportional to 1/j1+α where 0<α<1 the survival probability falls off as 1/nα. [ABSTRACT FROM AUTHOR]

Abstract: The passage of ions through membrane channels plays an important role in many fields of biology. An earlier paper [M. Boguñá, A.M. Berezhkovskii, G.H. Weiss, Phys. Rev. E 62 (2000) 3250] developed a toy model for statistical properties of the occupancy of a single site by different numbers of lattice random walkers chosen from an infinite set. It was assumed there that the residence time in one sojourn at the origin differed from the residence time of points elsewhere. In this paper we derive some properties of the corresponding first-passage time to the occupancy of the special site by random walkers in one or two dimensions. Results of our study were obtained from an extensive set of simulations. [Copyright &y& Elsevier]

In many biomedical applications of laser radiation to estimate optical parameters in tissue, it is desirable, if not crucial, to describe the region being probed. We suggest a parameter for this purpose, based on a lattice random walk on a simple cubic lattice. It is the expected number of distinct sites visited by a random walker. This is relatively easy to compute, although higher moments are not. Detailed computations are given for a continuous-wave measurement on a semi-infinite medium bounded by an absorbing plane and for a slab geometry required to analyse transillumination experiments. [ABSTRACT FROM AUTHOR]

It has been shown that many aspects of photon trajectories in tissue are well described by a lattice random walk model. We take advantage of this to characterize the region interrogated by photons in experiments designed to estimate internal optical parameters from light intensity re-emitted on the interface between tissue and the exterior in continuous-wave measurements. The parameter chosen to characterize the investigated region is the expected number of distinct sites visited by a photon that survives to reach the surface at a specific point. [Copyright &y& Elsevier]

A persistent random walk can be regarded as a multidimensional Markov process. The bias-free telegraphers equation is∂2p/∂t2+1/T∂p/∂t+1/T∂p/∂t ∂p/∂t=v2∇2p .It can be regarded as interpolating between the wave equation (T→∞) and the diffusion equation (T→0). Previously, it has found application in thermodynamics (cf. the review in Rev. Mod. Phys. 61 (1989) 41; 62 (1990) 375). More recent applications are reviewed in the present article. [Copyright &y& Elsevier]

An important question in the utilization of optical methods to estimate optical parameters non-invasively in human tissue is that of describing properties of the trajectories of laser-injected photons. It is relatively easy to calculate an expression for the average depth experienced by these photons starting from either a random-walk or a diffusion model. It is much harder to characterize more detailed descriptive parameters. Here we make use of a novel variation of the method of exact enumeration to provide essentially numerically exact results for the random-walk model of a continuous-wave experiment. [ABSTRACT FROM AUTHOR]

Reviews the approximate probabilistic approaches applied to crystallographic problems. Goals of modern crystallography; Background of crystallography; Ideal structure factor statistics; Orthogonal polynomial corrections of Non-ideal structure factor statistics; Exact representations of non-ideal structure; Factor statistics of multivariate structure; Other approximate techniques.

It is shown that a test suggested by Bechhofer, Kiefer, and Sobel [1], for selecting the better of two binomial populations, can be formulated and solved when a maximum number of tests is specified. If is assumed that the probability of correctly selecting the better population is specified to be better than P[sup *] when the difference in success probabilities exceeds a specified change[sup *]. The populations are sampled equally and it is assumed that the trial ends either when the difference in the number of successes exceeds a calculated value of s, or when N tests have been made, whichever is sooner. [ABSTRACT FROM AUTHOR]

In this article two sampling rules are analyzed for the problem of selecting the better of two binomial populations. The first is alternate sampling (vector-at-a. time) and the second is the "play-the-winner" rule introduced by Robbins [3], and discussed by Zelen [8]. The termination rule studied is inverse sampling. One result is that the probabilities of correct selection for these two methods of sampling are exactly equal. It is also found that the play-the-winner sampling is uniformly preferable to the vector-at-a-time sampling rule in the sense that for the same probability requirement the expected number of trials on the poorer population is always smaller. [ABSTRACT FROM AUTHOR]

There have been many studies of the merging delay problem; that is, the problem of calculating delay to a vehicle at a stop sign, waiting to merge with a single stream of traffic. In this article, a model is presented for merging from an acceleration lane of finite length. An expression for the expected delay is given. A driver in an acceleration lane will try to match his speed to the traffic on the major road, in order to create gaps that are as large as possible. Failing to merge while moving in the acceleration lane, the motorist will decelerate to come to a stop at the end of the lane. The calculations suggest that the expected delay does not depend strongly on the length of the acceleration lane and that most of the merging occurs near the beginning of the lane.

Results are developed for play-the-winner sampling for choosing the better of two binomial populations when the maximum number of tests is specified. It is shown that for a fixed number of tests the probability of correct selection with alternating assignment exceeds that for play- the-winner sampling. However, when the probability of correct selection is fixed, neither sampling method is uniformly better than the other as measured by the expected number of tests on the poorer population, or on the total expected number of tests. [ABSTRACT FROM AUTHOR]

The diffusion of molecules in biological tissues and some other microheterogeneous systems is affected by the presence of permeable barriers. This leads to the slowdown of diffusion at long times as compared to barrier-free diffusion. At short times the effect of barriers is weak. In consequence, the diffusion coefficient D(t) decreases as a function of time. We derive an exact solution for the Laplace transform of D(t) for diffusion in a space separated into layers by equally spaced, parallel identical planes of arbitrary permeability. Additionally, we give an approximation to D(t) which is reasonably accurate over the whole range of the partition permeability from zero (the case of isolated layers) to infinity (the case of no barriers). © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The theory of ligand binding to receptors on a cell surface suggested by Berg and Purcell and generalized by Zwanzig and Szabo uses the assumption that receptors are circular absorbing disks on an otherwise reflecting sphere. One of the key ingredients of this theory is a solution for the rate constant for ligand binding to a single circular receptor on a reflecting plane. We give an exact solution for the rate constant for binding to a single elliptic receptor and an approximate solution for binding to a single receptor of more general shape. The latter was tested by Brownian dynamics simulations. We found that the approximate formula predicted the rate constant with better than 10% accuracy for all studied receptor shapes. Using our solutions one can find the rate constant for ligand binding to a cell covered by N noncircular receptors by means of the Zwanzig–Szabo formula. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Analyzes the reaction dynamics on a thermally fluctuating potential. Jump rates for transitions between two states; Analysis of fast-fluctuation limit in terms of the potential of mean force; Turnover behavior.

Presents a letter to the editor regarding solution to an integral equation in the study of semi-poisson headway distribution.

Biomedical applications of near infrared radiation (NIR) techniques (i.e. based on light wavelengths roughly between 400 and 1100 nm) require that a preliminary estimate of the tissue volume being investigated be found. One possible estimate is the depth to which a photon penetrates a tissue before eventually emerging at a separating plane at a given time. A simple model for this problem can be based on a lattice random walk and was initially analyzed when the associated optical coefficients are isotropic with respect to the geometrical configuration. Here we include the effects of anisotropy in the optical coefficients, finding that at long times the statistical properties of the depth of penetration can be accounted for by very simple scaling factors while at short times the anisotropy effects can be quite noticeable. [ABSTRACT FROM AUTHOR]

We pose the problem of estimating the width of the upper layer of a two-stratum medium using data from reflectance measurements made at the interface separating the medium from its surroundings. The lower stratum is assumed to be infinite. In this analysis both the scattering and the absorption coefficients are allowed to differ, but are assumed to be known. We also suggest a simple way to include effects of noise into the analysis. [ABSTRACT FROM AUTHOR]

The number of distinct sites visited by a lattice random walk and its continuum analog, the volume swept out by a diffusing spherical particle are used to model different phenomena in physics, chemistry and biology. Therefore the problem of finding statistical properties of these random variables is of importance. There have been several studies of the more general problem of the volume of the region explored by N random walks or Brownian particles in an unbounded space. We here study the effects of a planar absorbing boundary on the average of this volume. The boundary breaks the translational invariance of the space, and introduces an additional spatial parameter, the initial distance of the Brownian particles from the surface. We derive expressions for the average volume visited in three dimensions and the average span in one dimension as functions of the time for given values of the initial distance to the absorbing boundary and N. The results can be transformed to those for N lattice random walks by appropriately choosing the radius and diffusion constant of the spheres. [Copyright &y& Elsevier]

We analyze the survival of a Brownian particle moving among randomly distributed spherical traps of two different radii. All of the results known in the case of identical traps, the mean-field Smoluchowski solution, the lowest order correction of this solution, the cumulant expansion, and the long-time behavior, are generalized to the case of two types of traps. Our approach is readily extended to deal with trapping spheres having more than two different radii. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

A problem common to biophysics, chemical engineering, physical chemistry, and physiology relates to a description of the kinetics of particle transport between two or more chambers. In this paper we analyze the case of two chambers connected by a cylindrical capillary. We derive general solutions for the Laplace transforms of the relaxation functions describing the equilibration of particles between the two chambers and the capillary. These solutions show how the relaxation functions depend on geometric parameters (volumes of the two chambers, the length and radius of the capillary) as well as diffusion coefficients in the three compartments. The general solutions are used to analyze the reduction to single-exponential kinetics which describes equilibration of the particles when the capillary is not too long. When all of the diffusion constants are equal we derive simple expressions for the average relaxation times. Brownian dynamics simulations were run to check the accuracy of approximations used to derive the results. We found excellent agreement between the theoretical predictions and numerical results. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The rate of a diffusion-controlled reaction with a buried binding site is smaller than the rate for the same site on the surface. We study the slowdown of the reaction rate when the site is hidden in a pore that connects two bulk media. On the assumption that the pore is cylindrical we derive an expression for the Laplace transform of the rate coefficient from which we infer the long-time limit of the reaction rate. This provides information on how the reaction rate depends on the channel radius, the location of the site, and the diffusion constant in the pore, which is allowed to differ from that in the bulk. The validity of approximations was checked by simulations that indicated excellent agreement between the analytical and and numerical results. [ABSTRACT FROM AUTHOR]

Abstract: We develop an analytic model of time-resolved fluorescent imaging of photons migrating through a semi-infinite turbid medium bounded by an infinite plane in the presence of a single stationary point fluorophore embedded in the medium. In contrast to earlier models of fluorescent imaging in which photon motion is assumed to be some form of continuous diffusion process, the present analysis is based on a continuous-time random walk (CTRW) on a simple cubic lattice, the objective being to estimate the position and lifetime of the fluorophore. This can provide information related to local variations in pH and temperature with potential medical significance. Aspects of the theory were tested using time-resolved measurements of the fluorescence from small inclusions inside tissue-like phantoms. The experimental results were found to be in good agreement with theoretical predictions provided that the fluorophore was not located too close to the planar boundary, a common problem in many diffusive systems. [ABSTRACT FROM AUTHOR]

Abstract: We study financial distributions within the framework of the continuous time random walk (CTRW). We review earlier approaches and present new results related to overnight effects as well as the generalization of the formalism which embodies a non-Markovian formulation of the CTRW aimed to account for correlated increments of the return. [Copyright &y& Elsevier]

Abstract: We adapt continuous time random walk (CTRW) formalism to describe asset price evolution and discuss some of the problems that can be treated using this approach. We basically focus on two aspects: (i) the derivation of the price distribution from high-frequency data, and (ii) the inverse problem, obtaining information on the market microstructure as reflected by high-frequency data knowing only the daily volatility. We apply the formalism to financial data to show that the CTRW offers alternative tools to deal with several complex issues of financial markets. [Copyright &y& Elsevier]