Montroll-Weiss Continuous-time Random Walk Research Articles

Asymmetric random walks with bias generated by discrete-time counting processes

We introduce a new class of asymmetric random walks on the one-dimensional infinite lattice. In this walk the direction of the jumps (positive or negative) is determined by a discrete-time renewal process which is independent of the jumps. We call this discrete-time counting process the ‘generator process’ of the walk. We refer the so defined walk to as ‘asymmetric discrete-time random walk’ (ADTRW). We highlight connections of the waiting time density generating functions with Bell polynomials. We derive the discrete-time renewal equations governing the time-evolution of the ADTRW and analyze recurrent/transient features of simple ADTRWs (walks with unit jumps in both directions). We explore the connections of the recurrence/transience with the bias: Transient simple ADTRWs are biased and vice verse. Recurrent simple ADTRWs are either unbiased in the large time limit or ‘strictly unbiased’ at all times with symmetric Bernoulli generator process. In this analysis we highlight the connections of bias and light-tailed/fat-tailed features of the waiting time density in the generator process. As a prototypical example with fat-tailed feature we consider the ADTRW with Sibuya distributed waiting times.We also introduce time-changed versions: We subordinate the ADTRW to a continuous-time renewal process which is independent from the generator process and the jumps to define the new class of ‘asymmetric continuous-time random walk’ (ACTRW). This new class – apart of some special cases – is not a Montroll–Weiss continuous-time random walk (CTRW). ADTRW and ACTRW models may open large interdisciplinary fields in anomalous transport, birth–death models and others.

Generalized Fractional Poisson Process and Related Stochastic Dynamics

Abstract We survey the ‘generalized fractional Poisson process’ (GFPP). The GFPP is a renewal process generalizing Laskin’s fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the ‘well-scaled’ diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.

Origins and applications of the Montroll-Weiss continuous time random walk

The Continuous Time Random Walk (CTRW) was introduced by Montroll and Weiss in 1965 in a purely mathematical paper. Its antecedents and later applications beginning in 1973 are discussed, especially for the case of fractal time where the mean waiting time between jumps is infinite.

On the non-equivalence of two standard random walks

We focus on two models of nearest-neighbour random walks on 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. 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.

Generalized master equation via aging continuous-time random walks.

We discuss the problem of the equivalence between continuous-time random walk (CTRW) and generalized master equation (GME). The walker, making instantaneous jumps from one site of the lattice to another, resides in each site for extended times. The sojourn times have a distribution density psi(t) that is assumed to be an inverse power law with the power index micro. We assume that the Onsager principle is fulfilled, and we use this assumption to establish a complete equivalence between GME and the Montroll-Weiss CTRW. We prove that this equivalence is confined to the case where psi(t) is an exponential. We argue that is so because the Montroll-Weiss CTRW, as recently proved by Barkai [E. Barkai, Phys. Rev. Lett. 90, 104101 (2003)], is nonstationary, thereby implying aging, while the Onsager principle is valid only in the case of fully aged systems. The case of a Poisson distribution of sojourn times is the only one with no aging associated to it, and consequently with no need to establish special initial conditions to fulfill the Onsager principle. We consider the case of a dichotomous fluctuation, and we prove that the Onsager principle is fulfilled for any form of regression to equilibrium provided that the stationary condition holds true. We set the stationary condition on both the CTRW and the GME, thereby creating a condition of total equivalence, regardless of the nature of the waiting-time distribution. As a consequence of this procedure we create a GME that is a bona fide master equation, in spite of being non-Markov. We note that the memory kernel of the GME affords information on the interaction between system of interest and its bath. The Poisson case yields a bath with infinitely fast fluctuations. We argue that departing from the Poisson form has the effect of creating a condition of infinite memory and that these results might be useful to shed light on the problem of how to unravel non-Markov quantum master equations.

A theory of 1/f current noise based on a random walk model

Asymptotic solutions for the Montroll-Weiss continuous-time random walk that are appropriate to the conduction of carriers through a resistive medium are utilized to show that, when carrier drift occurs by virtue of an applied electric field, a 1/f type of spectral density may be exhibited in the current noise. When the applied field is removed the spectral density is given by Nyquist's theorem.

Moments of the Montroll-Weiss continuous-time random walk for arbitrary starting time

The generalized version of the Montroll-Weiss formalism for continuous-time random walks is employed to show that some of the asymptotic results for large times appropriate to the ordinary walk become exact when the start of the observations is arbitrary.

On the relationship between continuous time random walks and the non-equilibrium Ornstein-Zernike equation

An exact non-equilibrium Ornstein-Zernike (OZ) equation is derived for lattice fluid systems whose time development is given by a generalized master equation. The derivation is based on a generalization of the Montroll-Weiss continuous-time random walk on a lattice, and on their relationship with master equation solutions. Time dependent direct and total correlation functions are defined in terms of the generating functions for the probability densities of the random walker, such that, in the infinite time limit the equilibrium OZ equation is recovered. A perturbative analysis of the time dependent OZ equation is shown to be formally analogous to the perturbation of the Bloch equation in quantum field theory. Analytic results are obtained, under the mean spherical approximation, for the time dependent total correlation function for a one-dimensional lattice fluid with exponential attraction.

Stochastic Transport in a Disordered Solid. I. Theory

A general theory of stochastic transport in disordered systems has been developed. The theory is based on a generalization of the Montroll-Weiss continuous-time random walk (CTRW) on a lattice. Starting from a general mobility formalism, specialized $\stackrel{\mathrm{\ifmmode\acute\else\textasciiacute\fi{}}}{\mathrm{t}}$o hopping conduction, an exact expression for the conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ for the CTRW process is derived. The frequency dependence of $\ensuremath{\sigma}(\ensuremath{\omega})$ is determined by the Fourier transform of the zeroth and second spatial moments of the function $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}},t)$, which is equal to the probability per unit time that the displacement and time between hops is $\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}$, $t$. The conductivity corresponding to characteristically different types of hopping distributions is discussed, as well as the basic approximation in adopting a CTRW on a lattice to transport in disordered solids.