Continuous-time random walks on networks with vertex- and time-dependent forcing (original) (raw)

Anomalous diffusion of self-propelled particles in directed random environments

Physical review. E, Statistical, nonlinear, and soft matter physics, 2014

We theoretically study the transport properties of self-propelled particles on complex structures, such as motor proteins on filament networks. A general master equation formalism is developed to investigate the persistent motion of individual random walkers, which enables us to identify the contributions of key parameters: the motor processivity, and the anisotropy and heterogeneity of the underlying network. We prove the existence of different dynamical regimes of anomalous motion, and that the crossover times between these regimes as well as the asymptotic diffusion coefficient can be increased by several orders of magnitude within biologically relevant control parameter ranges. In terms of motion in continuous space, the interplay between stepping strategy and persistency of the walker is established as a source of anomalous diffusion at short and intermediate time scales.

Random walk on graphs: An application to the double diffusivity model

Mechanics Research Communications, 2012

Preliminary but interesting and definite results are given on the application of graph theory concepts (random walk on graphs) to the double diffusivity theory proposed by Aifantis in the late 70s to model transport in media with high diffusivity paths such as metal polycrystals with a continuous distribution of grain boundaries possessing much higher diffusivity than the bulk, as well as in nanopolycrystals for which it has been shown recently that the double diffusivity model fits experimental observations. The new information provided by employing the graph theory tool is concerned with certain restrictions and relations that the phenomenological coefficients, entering in the coupled partial differential equations of double diffusivity, should satisfy depending on the topology and related details of the graph model adopted. © 2012 Elsevier Ltd. All rights reserved.

Continuous Time Random Walks with Reactions Forcing and Trapping

Mathematical Modelling of Natural Phenomena, 2013

One of the central results in Einstein's theory of Brownian motion is that the mean square displacement of a randomly moving Brownian particle scales linearly with time. Over the past few decades sophisticated experiments and data collection in numerous biological, physical and financial systems have revealed anomalous sub-diffusion in which the mean square displacement grows slower than linearly with time. A major theoretical challenge has been to derive the appropriate evolution equation for the probability density function of sub-diffusion taking into account further complications from force fields and reactions. Here we present a derivation of the generalised master equation for an ensemble of particles undergoing reactions whilst being subject to an external force field. From this general equation we show reductions to a range of well known special cases, including the fractional reaction diffusion equation and the fractional Fokker-Planck equation.

Scaling laws in the diffusion limited aggregation of persistent random walkers

Physica A: Statistical Mechanics and its Applications, 2011

We investigate the diffusion limited aggregation of particles executing persistent random walks. The scaling properties of both random walks and large aggregates are presented. The aggregates exhibit a crossover between ballistic and diffusion limited aggregation models. A non-trivial scaling relation ξ ∼ 1.25 between the characteristic size ξ, in which the cluster undergoes a morphological transition, and the persistence length , between ballistic and diffusive regimes of the random walk, is observed.

Global random walk modelling of transport in complex systems

Computing and Visualization in Science, 2007

The Global Random Walk algorithm performs simultaneously the tracking of large collections of particles and permits massive simulations at reasonable costs. Applications were developed for transport in systems with anisotropic, non-homogeneous, and randomly distributed parameters. As a first illustration we present simulations for diffusion in human skin. Further, a case study for contaminant transport in groundwater shows that the realizations of the transport process converge in mean square limit to a Gaussian diffusion. This investigation also indicates that the use of the Kraichnan routine, based on periodic random fields, yields reliable simulations of transport in Gaussian velocity fields.

Reaction-diffusion processes in scale-free networks

Noise in Complex Systems and Stochastic Dynamics, 2003

In this work we investigate the dynamics of reaction-diffusion processes on scale-free networks. Particles of two types, A and B, are randomly distributed on such a network and diffuse using random walk models by hopping to nearest neighbor nodes only. Here we treat the case where one species is immobile and the other is mobile. The immobile species acts as a trap, i.e. when particles of the other species encounter a trap node they are immediately annihilated. We numerically compute Φ(n, c), the survival probability of mobile species at time n, as a function of the concentration of trap nodes, c. We compare our results to the mean-field result (Rosenstock approximation), and the exact result for lattices of Donsker-Varadhan. We find that for high connectivity networks and high trap concentrations the mean-field result of a simple exponential decay is also valid here. But for low connectivity networks and low c the behavior is much more complicated. We explain these trends in terms of the number of sites visited, S n , the system size, and the concentration of traps.

Random collisions on branched networks: How simultaneous diffusion prevents encounters in inhomogeneous structures

Physical Review E, 2012

A huge variety of natural phenomena, including prey-predator interaction, chemical reaction kinetics, foraging, and pharmacokinetics, are mathematically described as encounters between entities performing a random motion on an appropriate structure. On homogeneous structures, two random walkers meet with certainty if and only if the structure is recurrent, i.e., a single random walker returns to its starting point with probability 1. We prove here that this property does not hold on general inhomogeneous structures, and introduce the concept of two-particle transience, providing examples of realistic recurrent structures where two particles may never meet if they both move, while an encounter is certain if either stays put. We anticipate that our results will pave the way for the study of the effects of geometry in a wide array of natural phenomena involving interaction between randomly moving agents.

All-time dynamics of continuous-time random walks on complex networks

The Journal of Chemical Physics, 2013

The concept of continuous-time random walks (CTRW) is a generalization of ordinary random walk models, and it is a powerful tool for investigating a broad spectrum of phenomena in natural, engineering, social and economic sciences. Recently, several theoretical approaches have been developed that allowed to analyze explicitly dynamics of CTRW at all times, which is critically important for understanding mechanisms of underlying phenomena. However, theoretical analysis has been done mostly for systems with a simple geometry. Here we extend the original method based on generalized master equations to analyze all-time dynamics of CTRW models on complex networks. Specific calculations are performed for models on lattices with branches and for models on coupled parallel-chain lattices. Exact expressions for velocities and dispersions are obtained. Generalized fluctuations theorems for CTRW models on complex networks are discussed.

On multiple-particle continuous-time random walks

Journal of Applied Mathematics, 2004

Scaling limits of continuous-time random walks are used in physics to model anomalous diffusion in which particles spread at a different rate than the classical Brownian motion. In this paper, we characterize the scaling limit of the average of multiple particles, independently moving as a continuous-time random walk. The limit is taken by increasing the number of particles and scaling from microscopic to macroscopic view. We show that the limit is independent of the order of these limiting procedures and can also be taken simultaneously in both procedures. Whereas the scaling limit of a single-particle movement has quite an obscure behavior, the multiple-particle analogue has much nicer properties.