Aggregation-fragmentation-diffusion model for trail dynamics

Abstract

We investigate statistical properties of trails formed by a random process incorporating aggregation, fragmentation, and diffusion. In this stochastic process, which takes place in one spatial dimension, two neighboring trails may combine to form a larger one, and also one trail may split into two. In addition, trails move diffusively. The model is defined by two parameters which quantify the fragmentation rate and the fragment size. In the long-time limit, the system reaches a steady state, and our focus is the limiting distribution of trail weights. We find that the density of trail weight has power-law tail P(w)∼w^{-γ} for small weight w. We obtain the exponent γ analytically and find that it varies continuously with the two model parameters. The exponent γ can be positive or negative, so that in one range of parameters small-weight trails are abundant and in the complementary range they are rare.

Lacunarity exponents

“…We show that there is a general mechanism explaining why the sparse regions can have a power-law distribution of density, and discuss examples where the exponent α of the low-density distribution, which we term the lacunarity exponent, can be estimated. In support of our claim that the effect is widely observable, we note that evidence which supports equation ( 1) has previously been presented in numerical and experimental studies of a few different systems with quite disparate equations of motion (the examples that we know about, [12][13][14][15], are discussed below). The term 'lacunarity' was introduced as a notion for characterising fractal sets by Mandelbrot [8], and various approaches to defining lacunarity have been explored [16,17], measuring the spatial inhomogeneity of a set (which need not be a fractal).…”

“…These observations concern models for the distribution of particles with quite different equations of motion, in different numbers of space dimensions. With the exception of [14], which uses an exact solution for a specific model, these papers do not offer a clear explanation for the power-law density distribution. This begs the question as to whether there is a common mechanism which implies power-law behaviour.…”

“…The exponential reduction of density for a single trajectory as a function of the number of iterations is a feature of all these systems: in most cases it is a result of the exponential separation of nearby trajectories, but in [14] it is the result of trajectories randomly splitting, reducing the weight in each daughter trajectory. In all of the examples in [12][13][14][15] except [13], the different trajectories can have multiple pre-images, and the probability of avoiding trajectories crossing or combining may be assumed to decrease exponentially as a function of time. The system considered by Larkin et al [13] involving particles floating on the surface of a liquid which is undergoing a turbulent or complex flow, is different: the equation of motion is ẋ = u(x, t), with a random velocity field u(x, t).…”

Numerical study of Bingham flow in macroscopic two dimensional heterogeneous porous media

“…It is important to note that this behavior results from the self-affine roughness of the flowing paths surrounding the cluster. For instance, if one would assume that a cluster is delimited by two paths from a random walk model, as proposed by [38] in another context, the cluster shape would then be described by a first-time return random walk. It would thus follow relationships like eqs.…”

Darcy law for yield stress fluid

“…For example, in [8,9] the statistics of the void was characterized numerically. Moreover, exact results have been obtained for the geometry of the branching paths of the first passage percolation model relevant for delta rivers [34] and animal trails [35].…”