Biased Continuous Time Random Walk Research Articles

Anomalous dielectric relaxation with multispecies linear reaction dynamics

Anomalous dielectric relaxation in complex reactions is difficult to specify, since there is no classic model for the reacting species with anomalous subdiffusion in the presence of an external electric field. In this paper, the anomalous dielectric relaxation of a multispecies system with linear reaction dynamics is studied based on the biased continuous time random walks. Different reaction–subdiffusion equations are derived via the mathematical random subordination time and the deterministic physical time. The mathematical random subordination time is based on the assumption that the jumping probabilities are determined at the end of the waiting times, while the deterministic physical time is based on the assumption that the jumping probabilities are independent of time and evaluated at the start of the waiting times. The force fields are operated by the Riemann–Liouville fractional derivative in the reaction–subdiffusion equations with the deterministic physical time, while the opposite results are found in the reaction–subdiffusion equations with the mathematical random subordination time. The explicit expressions for the anomalous dielectric relaxation of a reversible reaction are further derived according to the two reaction–subdiffusion equations in the presence of space- and time-dependent electric fields. The numerical calculations are carried to validate the reliability of the different expressions for the anomalous dielectric relaxation, and the reaction–subdiffusion equation with the deterministic physical time is proved to be suitable for describing the anomalous dielectric relaxation.

Squirrels can remember little: A random walk with jump reversals induced by a discrete-time renewal process

We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We derive exact formulae for the propagator using generating functions. We prove that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process. We consider the large-time asymptotics of the expected position: For waiting time densities with finite mean the walker remains in the average localized close to the departure site whereas escapes for fat-tailed waiting-time densities (i.e. densities with infinite mean) by a sublinear power-law. We explore anomalous diffusion features by accounting for the ‘aging effect’ as a hallmark of non-Markovianity where the discrete-time version of the ‘aging renewal process’ comes into play. By deriving pertinent distributions of this process we obtain explicit formulae for the variance when the waiting-times are Sibuya-distributed. In this case and generally for fat-tailed waiting time PDFs a t2-ballistic superdiffusive scaling emerges in the large time limit. In contrast if the waiting time PDF between the step reversals is light-tailed (‘narrow’ with finite mean and variance) the walk exhibits normal diffusion and for ‘broad’ waiting time PDFs (with finite mean and infinite variance) superdiffusive large time scaling. We also consider time-changed versions where the walk is subordinated to a continuous-time point process such as the time-fractional Poisson process. This defines a new class of biased continuous-time random walks exhibiting several regimes of anomalous diffusion.

Active random walks in one and two dimensions.

We investigate active lattice walks: biased continuous time random walks which perform orientational diffusion between lattice directions in one and two spatial dimensions. We study the occupation probability of an arbitrary site on the lattice in one and two dimensions and derive exact results in the continuum limit. Next, we compute the large deviation free-energy function in both one and two dimensions, which we use to compute the moments and the cumulants of the displacements exactly at late times. Our exact results demonstrate that the cross-correlations between the motion in the x and y directions in two dimensions persist in the large deviation function. We also demonstrate that the large deviation function of an active particle with diffusion displays two regimes, with differing diffusive behaviors. We verify our analytic results with kinetic Monte Carlo simulations of an active lattice walker in one and two dimensions.

Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

Reaction–subdiffusion equations for the [formula omitted] reaction in space- and time-dependent force fields: A study for the anomalous dielectric relaxation

To describe the anomalous dielectric relaxation of reaction in space- and time-dependent electric field, we derive the possible kinetic equations for describing the distributions of A and B particles in a simple reaction A→B based on the biased continuous time random walk, when the jumping probabilities are evaluated after or before the waiting time, respectively. Different reaction—subdiffusion equations are derived for different cases, in which the properties of A and B particles are the same or different. The reaction and subdiffusion terms can be decoupled if the past is forgotten when B particles are converted by A particles. According to the corresponding kinetic equations in the spherical coordinates, the dielectric polarization of the reaction is derived. By comparing the different expressions of the polarization, it is found that the reaction–subdiffusion equation based on the jumping probabilities, which are independent of time and evaluated before the waiting time, is more reasonable to describe the dielectric relaxation process in the chemical reaction.

Transport in disordered systems: The single big jump approach

In a growing number of strongly disordered and dense systems, the dynamics of a particle pulled by an external force field exhibits super-diffusion. In the context of glass forming systems, super cooled glasses and contamination spreading in porous medium it was suggested to model this behavior with a biased continuous time random walk. Here we analyze the plume of particles far lagging behind the mean, with the single big jump principle. Revealing the mechanism of the anomaly, we show how a single trapping time, the largest one, is responsible for the rare fluctuations in the system. These non typical fluctuations still control the behavior of the mean square displacement, which is the most basic quantifier of the dynamics in many experimental setups. We show how the initial conditions, describing either stationary state or non-equilibrium case, persist for ever in the sense that the rare fluctuations are sensitive to the initial preparation. To describe the fluctuations of the largest trapping time, we modify Fr\'{e}chet's law from extreme value statistics, taking into consideration the fact that the large fluctuations are very different from those observed for independent and identically distributed random variables.

Ergodicity, rejuvenation, enhancement, and slow relaxation of diffusion in biased continuous-time random walks.

Bias plays an important role in the enhancement of diffusion in periodic potentials. Using the continuous-time random walk in the presence of a bias, we report on an interesting phenomenon for the enhancement of diffusion by the start of the measurement in a random energy landscape. When the variance of the waiting time diverges, in contrast to the bias-free case, the dynamics with bias becomes superdiffusive. In the superdiffusive regime, we find a distinct initial ensemble dependence of the diffusivity. Moreover, the diffusivity can be increased by the aging time when the initial ensemble is not in equilibrium. We show that the time-averaged variance converges to the corresponding ensemble-averaged variance; i.e., ergodicity is preserved. However, trajectory-to-trajectory fluctuations of the time-averaged variance decay unexpectedly slowly. Our findings provide a rejuvenation phenomenon in the superdiffusive regime, that is, the diffusivity for a nonequilibrium initial ensemble gradually increases to that for an equilibrium ensemble when the start of the measurement is delayed.

Impact of diffusive motion on anomalous dispersion in structured disordered media: From correlated Lévy flights to continuous time random walks.

We elucidate the impact of diffusive motion on the nature of anomalous dispersion in layered and fibrous disordered media. We consider two types of disorder characterized by quenched random velocities and quenched random retardation properties. Purely advective particle motion is ballistic in both disorder models. This changes dramatically in the presence of transverse diffusion, which leads to dimension-dependent disorder sampling. For d≤3 dimensions, heavy-tailed velocity distributions render large-scale particle motion a correlated Lévy flight, while transport in the quenched random retardation model behaves as a biased continuous time random walk with correlated time increments.

Biased continuous-time random walks for ordinary and equilibrium cases: facilitation of diffusion, ergodicity breaking and ageing.

We examine renewal processes with power-law waiting time distributions (WTDs) and non-zero drift via computing analytically and by computer simulations their ensemble and time averaged spreading characteristics. All possible values of the scaling exponent α are considered for the WTD ψ(t) ∼ 1/t1+α. We treat continuous-time random walks (CTRWs) with 0 < α < 1 for which the mean waiting time diverges, and investigate the behaviour of the process for both ordinary and equilibrium CTRWs for 1 < α < 2 and α > 2. We demonstrate that in the presence of a drift CTRWs with α < 1 are ageing and non-ergodic in the sense of the non-equivalence of their ensemble and time averaged displacement characteristics in the limit of lag times much shorter than the trajectory length. In the sense of the equivalence of ensemble and time averages, CTRW processes with 1 < α < 2 are ergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW renewal processes with α > 2-both for the equilibrium and ordinary situation-are always ergodic. For the situations 1 < α < 2 and α > 2 the variance of the diffusion process, however, depends on the initial ensemble. For biased CTRWs with α > 1 we also investigate the behaviour of the ergodicity breaking parameter. In addition, we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements, in the entire range of the WTD exponent α.

Occupation times and ergodicity breaking in biased continuous time randomwalks

Continuous time random walk (CTRW) models are widely used to model diffusion incondensed matter. There are two classes of such models, distinguished by the convergenceor divergence of the mean waiting time. Systems with finite average sojourn time areergodic and thus Boltzmann–Gibbs statistics can be applied. We investigate the statisticalproperties of CTRW models with infinite average sojourn time; in particular, theoccupation time probability density function is obtained. It is shown that in thenon-ergodic phase the distribution of the occupation time of the particle on a given latticepoint exhibits bimodal U or trimodal W shape, related to the arcsine law. The key pointsare as follows. (a) In a CTRW with finite or infinite mean waiting time, the distribution ofthe number of visits on a lattice point is determined by the probability that a member ofan ensemble of particles in equilibrium occupies the lattice point. (b) The asymmetryparameter of the probability distribution function of occupation times is related to theBoltzmann probability and to the partition function. (c) The ensemble average is givenby Boltzmann–Gibbs statistics for either finite or infinite mean sojourn time,when detailed balance conditions hold. (d) A non-ergodic generalization of theBoltzmann–Gibbs statistical mechanics for systems with infinite mean sojourn time isfound.

Multipoint correlation functions for continuous-time random walk models of anomalous diffusion

Recursive relations are developed for computing the multipoint correlation functions of a particle undergoing a biased continuous-time random walk (CTRW) in an external potential. Two- and three-point correlation functions are calculated for waiting-time distributions with an anomalous power-law profile t(-alpha-1), 0 < alpha < 1, on intermediate time scales with a crossover to an exponential long time decay. Comparison of the CTRW with the Brownian harmonic oscillator model (Gaussian process) illustrates how higher-order correlation functions may be used to distinguish between dynamical models that have the same two-point correlation function.

FIRST PASSAGE TIME PROBLEM FOR BIASED CONTINUOUS-TIME RANDOM WALKS

We study the first passage time (FPT) problem for biased continuous time random walks. Using the recently formulated framework of fractional Fokker-Planck equations, we obtain the Laplace transform of the FPT density function when the bias is constant. When the bias depends linearly on the position, the full FPT density function is derived in terms of Hermite polynomials and generalized Mittag-Leffler functions.

Vacancy diffusion in colloidal crystals

The diffusion of vacancies in a one-dimensional colloidal crystal has been studied. The vacancy diffusion was constructed by superposition of single-step hopping processes using the framework of a biased continuous-time random walk. The statistics of single-step hopping between neighbouring lattice sites have been evaluated using an accurate numerical scheme for solving the Fokker–Planck equation. The model was used to explain the stratification mechanism of thin liquid films formed from macroparticle suspensions.