Energy transfer as a random walk with long-range steps

We consider the incoherent energy transport in molecular crystals, where the transfer rates stem from Coulombic and exchange interactions. For substitutionally disordered lattices we present in a first passage model the excitation decay due to trapping by randomly distributed traps; the decay is related to the distribution of the number of distinct sites visited during the timet and is expressible through the cumulants of this distribution. The validity domains of approximate decay laws based on the first few cumulants are also discussed. We exemplify the findings for dipolar transfer rates between randomly distributed molecules on a square lattice, by comparing the random walk on the random system to its CTRW (continuous time random walk) counterpart.     

Lattice random walks for sets of random walkers. First passage times

We have studied the mean first passage time for the first of aset of random walkers to reach a given lattice point on infinite lattices ofD dimensions. In contrast to the well-known result ofinfinite mean first passage times for one random walker in all dimensionsD, we findfinite mean first passage times for certain well-specified sets of random walkers in all dimensions, exceptD = 2. The number of walkers required to achieve a finite mean time for the first walker to reach the given lattice point is a function of the lattice dimensionD. ForD > 4, we find that only one random walker is required to yield a finite first passage time, provided that this random walker reaches the given lattice point with unit probability. We have thus found a simple random walk property which sticks atD > 4.Supported in part by a grant from Charles and Renée Taubman and by the National Science Foundation, Grant CHE78-21460.     

First passage time and escape time distributions for continuous time random walks

We consider an arbitrary continuous time random walk (ctrw)via unbiased nearest-neighbour jumps on a linear lattice. Solutions are presented for the distributions of the first passage time and the time of escape from a bounded region. A simple relation between the conditional probability function and the first passage time distribution is analysed. So is the structure of the relation between the characteristic functions of the first passage time and escape time distributions. The mean first passage time is shown to diverge for all (unbiased)ctrw’s. The divergence of the mean escape time is related to that of the mean time between jumps. A class ofctrw’s displaying a self-similar clustering behaviour in time is considered. The exponent characterising the divergence of the mean escape time is shown to be (1−H), whereH(0<H<1) is the fractal dimensionality of thectrw.     

Residence Time Statistics for Normal and Fractional Diffusion in a Force Field

We investigate statistics of occupation times for an over-damped Brownian particle in an external force field, using a backward Fokker–Planck equation introduced by Majumdar and Comtet. For an arbitrary potential field the distribution of occupation times is expressed in terms of solutions of the corresponding first passage time problem. This general relationship between occupation times and first passage times, is valid for normal Markovian diffusion and for non-Markovian sub-diffusion, the latter modeled using the fractional Fokker–Planck equation. For binding potential fields we find in the long time limit ergodic behavior for normal diffusion, while for the fractional framework weak ergodicity breaking is found, in agreement with previous results of Bel and Barkai on the continuous time random walk on a lattice. For non-binding cases, rich physical behaviors are obtained, and classification of occupation time statistics is made possible according to whether or not the underlying random walk is recurrent and the averaged first return time to the origin is finite. Our work establishes a link between fractional calculus and ergodicity breaking.     

Random walks on the Bethe lattice

We obtain random walk statistics for a nearest-neighbor (Pólya) walk on a Bethe lattice (infinite Cayley tree) of coordination numberz, and show how a random walk problem for a particular inhomogeneous Bethe lattice may be solved exactly. We question the common assertion that the Bethe lattice is an infinite-dimensional system.Supported in part by the U.S. Department of Energy.     

Random walks with nonnearest-neighbor transitions. II. Analytic one-dimensional theory for exponentially distributed steps in systems with boundaries

Exact analytic results for symmetric, nonnearest-neighbor random walks in one-dimensional finite and semiinfinite lattices are presented. Random walks with exponentially distributed step lengths are considered such that variation of a single parameter permits one to cover the whole range of step lengths from nearest-neighbor transitions to steps of aribtrary length. The generating functions for such lattices are derived and used to calculate a number of moment properties (mean first passage times, dispersion in the mean recurrence time). Since explicit expressions for the generating functions for these walks are obtained, additional moment properties can readily be calculated. The results found here for a finite system are compared to results found previously for a system with periodic boundary conditions. Two different semiinfinite systems are also considered.     

Mean first passage time for a class of non-Markovian processes

We determine the probability distribution of the first passage time for a class of non-Markovian processes. This class contains, amongst others, the well-known continuous time random walk (CTRW), which is able to account for many properties of anomalous diffusion processes. In particular, we obtain the mean first passage time for CTRW processes with truncated power-law transition time distribution. Our treatment is based on the fact that the solutions of the non-Markovian master equation can be obtained via an integral transform from a Markovian Langevin process.     

First passage time distributions for finite one-dimensional random walks

We present closed expressions for the characteristic function of the first passage time distribution for biased and unbiased random walks on finite chains and continuous segments with reflecting boundary conditions. Earlier results on mean first passage times for one-dimensional random walks emerge as special cases. The divergences that result as the boundary is moved out to infinity are exhibited explicitly. For a symmetric random walk on a line, the distribution is an elliptic theta function that goes over into the known Lévy distribution with exponent 1/2 as the boundary tends to ∞.     

First passage time problems for a class of master equations with separable kernels

We discuss first passage time problems for a class of one-dimensional master equations with separable kernels. For this class of master equations the integral equation for first passage time moments can be transformed exactly into ordinary differential equations. When the separable kernel has only a single term the equation for the mean first passage time obtained is exactly that for simple diffusion. The boundary conditions, however, differ from those appropriate to simple diffusion. The equations for higher moments differ slightly from those for simple diffusion. Analysis is presented, of a generalization of a model of a random walk with long-range jumps first investigated by Lindenberg and Shuler. Since the equations can be solved exactly one can study the behavior of boundary conditions in the continuum limit. The generalization to a larger number of terms in the separable kernel leads to higher order equations for the first passage time moments. In each case, boundary conditions can be found directly from the original master equation.     

A kinetic model for the premelting of a crystalline structure

An analytical kinetic approach to examine the premelting phenomenon is suggested by using a first passage time analysis. Premelting is considered to occur when the time of formation of a Frenkel type defect in the surface monolayer becomes sufficiently small. The mean time of defect formation on the surface lattice, i.e., the mean time necessary for a selected (surface-located) molecule to leave its lattice site and form a Frenkel defect, is calculated by using a first passage time analysis. The model is illustrated by numerical calculations for a crystalline structure composed of molecules interacting via the Lennard-Jones (LJ) potential. The lattice vectors in the plane parallel to the free surface of the crystal were assumed to be equal (to the lattice parameter) and the angle between them was varied. The model predictions of the Tammann temperature (of premelting) are very sensitive to the parameters of the LJ potential. In all the cases considered, the temperature dependence of the mean first passage time has two clearly distinct regimes: at low temperatures the dependence is sharp and at high temperatures it is weak.     

Inhomogeneous random sequential adsorption on a lattice

We study idealized random sequential adsorption on a lattice, with adsorption probabilities inhomogeneous both in space and in time, and including the possibility of cooperativity. Attention is directed to the mean occupancy of a given site as a function of time, which is represented by a weighted random walk on the lattice. In the special case of nearest neighbor exclusion, the walk is transformed to one in which only neighbors of occupied sites can be occupied, but with a renormalized probability. Reduction theorems are presented, with which the general case of a tree lattice is completely solved in inverse form.     

Random walks on sparsely periodic and random lattices: I. Random walk properties from lattice bond enumeration

We have developed a new technique for calculating certain asymptotic random walk properties on sparsely periodic and related random lattices in two and three dimensions. This technique is based on an ansatz which relates the number of lattice bonds in “irreducible lattice fragments” to the number of steps along these bonds. We show that certain random walk properties can be calculated very simply on the basis of this ansatz and that they depend only on the density of bonds and not on the arrangement of the bonds within the lattice. The random walk properties calculated here (mean square displacements, number of distinct sites visited, probability of return to the origin) are in complete agreement with results obtained earlier via generating function techniques. A subsequent paper contains generating function calculations which verify a number of new results presented here, such as mean occupation frequency of lattice sites, and a proof of our basic assumption on the relation between the number of lattice bonds and random walk steps.     

A lower bound for the mass of a random Gaussian lattice

We give a criterion that the two point function for a Gaussian lattice with random mass decay exponentially. The proof uses a random walk representation which may be of interest in other contexts.Supported by N.S.F. Grants PHY 76-17191, MPS 10751     

Random walks with infinite spatial and temporal moments

The continuous-time random walk of Montroll and Weiss has been modified by Scher and Lax to include a coupled spatial-temporal memory. We treat novel cases for the random walk and the corresponding generalized master equation when combinations of both spatial, and temporal moments of the memory are infinite. The asymptotic properties of the probability distribution for being at any lattice site as a function of time and its variance are calculated. The resulting behavior includes localized, diffusive, wavelike, and Levy's stable laws for the appropriate scaled variable. We show that an infinite mean waiting time can lead to long time diffusive behavior, while a finite mean waiting time is not sufficient to ensure the same.     

A generalized simple random walk in one dimension related to the Gaussian polynomials

A generalization of the relation between the simple random walk on a regular lattice and the diffusion equation in a continuous space is described. In one dimension we consider a random walk of a walker with exponentially decreasing mobility with respect to time. It has an exact solution of the conditional probability, that is expressed in terms of the Gaussian polynomials, a generalization of binomial coefficients. Taking a suitable continuum limit we obtain the corresponding transport equation from the recursion relation of the discrete random walk process. The kernel of this differential equation is also directly obtained from that conditional probability by the same continuum limit.     

On the non-equivalence of two standard random walks

We focus on two models of nearest-neighbour random walks on d$d$-dimensional regular hyper-cubic lattices that are usually assumed to be identical—the discrete-time Polya walk, in which the walker steps at each integer moment of time, and the Montroll–Weiss continuous-time random walk in which the time intervals between successive steps are independent, exponentially and identically distributed random variables with mean 1$1$. We show that while for symmetric random walks both models indeed lead to identical behaviour in the long time limit, when there is an external bias they lead to markedly different behaviour.     

Lévy Walk with Multiple Internal States

Lévy walk with multiple internal states can effectively model the motion of particles that don’t immediately move back to the directions or areas which they come from. When the Lévy walk behaves superdiffusion, it is discovered that the non-immediately-repeating property, characterized by the constructed transition matrix, has no influence on the particle’s mean square displacement (MSD) or Pearson coefficient. This is a kind of stable property of Lévy walk. However, if the Lévy walk shows the dynamical behaviors of normal diffusion, then the effect of non-immediately-repeating emerges. For the Lévy walk with some particular transition matrices, it may display nonsymmetric dynamics; in these cases, the behaviors of their variances are detailedly discussed, especially some comparisons with the ones of the continuous time random walks are made (a striking difference is the changes of the exponents of the variances). The first passage time distribution and its average of Lévy walks are simulated, the results of which turn out that the first passage time can distinguish Lévy walks with different transition matrices, while the MSD can not.

     

Hopping on a zig-zag course

We study the diffusion coefficient of Active Brownian particles in two dimensions. In addition to usual attributes of active motion we let the particles turn in preferred directions over random times. This angular motion is modeled by an effective Lorentz force with time dependent frequency switching between two values at exponentially distributed random times. The diffusion coefficient is calculated by the Taylor-Kubo formula where distributions found from a Fokker-Planck equation or from a continuous time random walk approach have been inserted for averaging. Eventually properties of the diffusion coefficient will be discussed.     

First passage time densities for random walk spans

A general expression is derived for the Laplace transform of the probability density of the first passage time for the span of a symmetric continuous-time random walk to reach levelS. We show that when the mean time between steps is finite, the mean first passage time toS is proportional toS ₂. When the pausing time density is asymptotic to a stable density we show that the first passage density is also asymptotically stable. Finally when the jump distribution of the random walk has the asymptotic formp(j)A/|j| ₊₁, 0 < < 2 it is shown that the mean first passage time toS goes likeS .     

Fractional nonlinear diffusion equation and first passage time

We investigate the solutions and the first passage time for anomalous diffusion processes governed by the fractional nonlinear diffusion equation with a space- and time-dependent diffusion coefficient subject to absorbing boundaries and the initial condition. We obtain explicit analytical expression for the probability distribution, the first passage time distribution, the mean first passage time, and the mean squared displacement corresponding to different time-dependent diffusion coefficient. In addition, we compare our results for the first passage time distribution and the mean first passage time with the one obtained by usual linear diffusion equation with time-dependent diffusion coefficient.