Lattice Random Walk Models Research Articles

Fluctuation relation in continuous-time random walks driven by an external field

We study a fluctuation relation representing a non-equilibrium equality indicating that the ratio between the distribution of trajectories obtained by exchanging the initial and final positions is characterized by free energy differences for the duration of the trajectories. We examine the fluctuation relation for noninteracting charge carriers driven by an external electric field by using a continuous-time lattice random walk model with a general waiting-time distribution of transitions. The fluctuation relation is obtained regardless of the lattice structure factor or the form of the waiting-time distribution. However, the fluctuation relation is satisfied only after taking the continuum limit in the presence of a reflecting boundary. Moreover, in free space without boundary conditions, exchanging the initial and final positions is equivalent to exchanging the field (or drift) directions. However, we show that the exchanging field (or drift) directions is not relevant for studying the fluctuation relation under the reflecting boundary condition.

Diffusion through permeable interfaces: Fundamental equations and their application to first-passage and local time statistics

The diffusion equation is the primary tool to study the movement dynamics of a free Brownian particle, but when spatial heterogeneities in the form of permeable interfaces are present, no fundamental equation has been derived. Here we obtain such an equation from a microscopic description using a lattice random walk model. The sought after Fokker-Planck description and the corresponding backward Kolmogorov equation are employed to investigate first-passage and local time statistics and gain new insights. Among them a surprising phenomenon, in the case of a semibounded domain, is the appearance of a regime of dependence and independence on the location of the permeable barrier in the mean first-passage time. The new formalism is completely general: it allows to study the dynamics in the presence of multiple permeable barriers as well as reactive heterogeneities in bounded or unbounded domains and under the influence of external forces.

Anderson-like localization transition of random walks with resetting

We study several lattice random walk models with stochastic resetting to previously visited sites which exhibit a phase transition between an anomalous diffusive regime and a localization regime where diffusion is suppressed. The localized phase settles above a critical resetting rate, or rate of memory use, and the probability density asymptotically adopts in this regime a non-equilibrium steady state similar to that of the well known problem of diffusion with resetting to the origin. The transition occurs because of the presence of a single impurity site where the resetting rate is lower than on other sites, and around which the walker spontaneously localizes. Near criticality, the localization length diverges with a critical exponent that falls in the same class as the self-consistent theory of Anderson localization of waves in random media. The critical dimensions are also the same in both problems. Our study provides analytically tractable examples of localization transitions in path-dependent, reinforced stochastic processes, which can also be useful for understanding spatial learning by living organisms.

Random walks in nonuniform environments with local dynamic interactions

We consider a class of lattice random walk models in which the random walker is initially confined to a finite connected set of allowed sites but has the opportunity to enlarge this set by colliding with its boundaries, each such collision having a given probability of breaking through. The model is motivated by an analogy to cell motility in tissue, where motile cells have the ability to remodel extracellular matrix, but is presented here as a generic model for stochastic erosion. For the one-dimensional case, we report some exact analytic results, some mean-field type analytic approximate results and simulations. We compute exactly the mean and variance of the time taken to enlarge the interval from a single site to a given size. The problem of determining the statistics of the interval length and the walker's position at a given time is more difficult and we report several interesting observations from simulations. Our simulations include the case in which the initial interval length is random and the case in which the initial state of the lattice is a random mixture of allowed and forbidden sites, with the walker placed at random on an allowed site. To illustrate the extension of these ideas to higher-dimensional systems, we consider the erosion of the simple cubic lattice commencing from a single site and report simulations of measures of cluster size and shape and the mean-square displacement of the walker.

A new set of Monte Carlo moves for lattice random-walk models of biased diffusion

We recently demonstrated that standard fixed-time lattice random-walk models cannot be modified to represent properly the biased diffusion processes in more than two dimensions. The origin of this fundamental limitation appears to be the fact that traditional Monte Carlo moves do not allow for simultaneous jumps along each spatial direction. We thus propose a new algorithm to transform biased diffusion problems into lattice random walks such that we recover the proper dynamics for any number of spatial dimensions and for arbitrary values of the external field. Using a hypercubic lattice, we redefine the basic Monte Carlo moves, including the transition probabilities and the corresponding time durations, in order to allow simultaneous jumps along all Cartesian axes. We show that our new algorithm can be used both with computer simulations and with exact numerical methods to obtain the mean velocity and the diffusion coefficient of point-like particles in any dimension and in the presence of obstacles.

Exact lattice calculations of dispersion coefficients in the presence of external fields and obstacles.

We present a study of the field-dependent dispersion coefficient of point-like particles in various 2D overdamped systems with obstructions (periodic, percolating, and trapping distributions of obstacles). These calculations profit from the synthesis of a newly proposed Monte Carlo algorithm--the first such algorithm that correctly reproduces the free dispersion coefficient in the presence of finite external fields--and an asymptotically exact calculation technique. The resulting method efficiently produces algebraic and numerical results without the need to actually perform Monte Carlo simulations. When compared to such simulations, our exact method features a negligible computational cost and exponentially small errors. Utilizing the power of this numerical method, we engage in comprehensive parametric analysis of several model systems, revealing very subtle effects that would otherwise be swamped by statistical errors or incur prohibitive computational costs. The unified framework presented here serves as a template for further applications of lattice random-walk models of biased diffusion.

Building reliable lattice Monte Carlo models for real drift and diffusion problems.

We revisit the well-known issue of representing an overdamped drift-and-diffusion system by an equivalent lattice random-walk model. We demonstrate that commonly used Monte Carlo algorithms do not conserve the diffusion coefficient when a driving field of arbitrary amplitude is present, and that such algorithms would actually require fluctuating jumping times and one clock per Cartesian direction to work properly. Although it is in principle possible to construct valid algorithms with fixed time steps, we show that no such algorithm can be used in more than two dimensions if the jumps are made along only one axis at each time step.

Absorptive effects on a parameter used to characterize the region of tissue visited by photons in continuous wave measurements

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.

First-order coil-to-flower transition of a polymer chain pinned near a stepwise external potential: Numerical, analytical, and scaling analysis

A polymer chain near a penetrable interface is studied in the Gaussian model, in the lattice random walk model and by a scaling analysis. The interface is modeled as an external potential u of a Heaviside step-function form. One end of the chain is fixed at a distance z0 away from this interface. When the end point is fixed in the high potential region, a first-order coil-to-flower transition takes place upon variation of the distance z0. Here, the flower has a strongly stretched stem from the grafting point towards the interface and, on top of it, a crown composed of the remaining segments in a (perturbed) coil conformation. The coil-to-flower transition is analyzed in terms of the Landau free energy. The order parameter is taken to be related to the fraction of segments residing in the energetically favorable region. Exact analytical expressions for the Landau function are obtained in the Gaussian model for any distances z0 and potential strength u. A phase diagram in the z0 versus u coordinates is constructed. It contains a line of the first-order phase transitions (binodal line) ending at a critical point z0=u=0, and two spinodal lines. Numerical results are obtained for several chain lengths in the lattice random walk model demonstrating the effects of finite extensibility on the position of the transition point. Excluded volume effects are analyzed within the scaling approach.

A measure of photon penetration into tissue in diffusion models

In application of laser-based techniques to estimate optical parameters of tissue it is necessary to know the parts of the tissue that have been sampled by the photons. Towards this end it is desirable to define an average depth of a photon trajectory. This has been done earlier for lattice random walk models of photon trajectories where one simply counts the number of visits to a given set of sites. Here we calculate the equivalent quantity in the diffusion picture by equating the average depth of penetration to the local time of a diffusion process. We show that for a semi-infinite medium bounded by a plane the average depth probed by a photon trajectory in a continuous-wave (cw) experiment has the same functional form as that calculated from a random walk model. The local time, or average depth is shown to take a very simple form in the case of a time-gated experiment. We extend our results to the cw transillumination experiment.

The continuous-time random walk description of photon motion in an isotropic medium

Lattice random walk models are useful alternatives to diffusion theory for describing photon migration in turbid media. Results obtained from random walk models were, in the past, based on approximating the propagator as a Gaussian function. This is equivalent to assuming that a large number of steps have been taken. We show that a particular, physically plausible, continuous-time random walk (CTRW) model can be solved exactly. At sufficiently long times the solution reduces to that for a random walk in discrete time. We provide exact solutions to problems related to cw reflection measurements and indicate some generalizations to time-gated measurements made on a uniform medium.

Absorptivity contrast in transillumination imaging of tissue abnormalities

We calculate the time-resolved flux of photons transmitted across an optically turbid slab containing a partially absorbing inclusion. An analytical expression is obtained for the flux at a detector positioned opposite a point source (at a distance equal to the thickness of the slab) when the center of the inclusion lies on the line connecting those points. The calculation employs a discrete-time lattice random-walk model of photon transport. The resulting expression is used to assess the affects of time resolution on the detectability of the inclusion.

Statistics of Penetration Depth of Photons Re-emitted from Irradiated Tissue

Abstract We derive some statistical properties of photons emitted from bulk tissue that has been irradiated by a continuous beam of laser light. The properties include the distribution and moments of the depth reached by a photon, conditional on its having reached the surface at a (dimensionless) distance, ϱ, from the point of entry of the beam into the tissue. Theoretical predictions are made on the basis of a lattice random-walk model and are checked by simulations of the light penetration problem for a more realistic continuum model.

EXCITON MIGRATION AND TRAPPING IN PHOTOSYNTHESIS

Abstract— Mobile electronic excited states, excitons, undergo random walks through the antenna chlorophyll arrays of photosynthetic organisms. The time interval from exciton creation, by photon absorption, until its first arrival at a reaction center (RC) is called the “first passage time” (FPT) of the random walk. A theory of exciton migration and trapping presented here predicts that the exciton lifetime, as measured from chlorophyll fluorescence decay in chromatophores or P700 complexes, is a linear function of the fractional number of quanta absorbed directly by the antenna, not by the RC. The slope of this line is the FPT, and its intercept is the exciton's lifetime as limited only by photoconversion at the RC. This photoconversion‐limited lifetime is simply related to the in situ photoconversion rate constant via two parameters, each of which is experimentally accessible. It is also possible to obtain values of individual FoUrster rate constants, at least approximately, from measurements of exciton lifetime as functions of temperature and excitation wavelength. This new theory, based on lattice random walk models, receives some support from fluorescence measurements done on Rhodopseudomonas sphaeroides R26 chromatophores. In its present form the theory is only applicable to one‐antenna‐component systems, like Rp. sphaeroides R26 or Rhodospirillum rubrum chromatophores or P700 complexes, but should be readily extendible to multi‐antenna‐component systems including whole chloroplasts.

A comparison between real polymer dimensions in solution and the behaviour predicted from models

The variation of polymer coil dimensions with numbers of bonds in the chain for real polymers in solution is compared with the behaviour predicted by lattice random walk models. There appears to be agreement between one form, predicted by two independent methods from models, and the real behaviour of polymers.