Engineering topics

Ben-Naim, Eli

Publications and source records attributed to Ben-Naim, Eli.

8 records

DOE OSTI2022

Statistical properties of sites visited by independent random walks

The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. Here, we consider two independent random walks on hyper-cubic lattices and study ordering probabilities associated with these characteristics. The first is the probability that during the time interval (0, t), the number of sites visited by a walker never exceeds that of another walker. The second is the probability that the sites visited by a walker remain a subset of the sites visited by another walker. Using numerical simulations, we investigate the leading asymptotic behaviors of the ordering probabilities in spatial dimensions d = 1, 2, 3, 4. We also study the time evolution of the number of ties between the number of visited sites. We show analytically that the average number of ties increases as a 1 ln t with a 1 = 0.970 508 in one dimension and as (ln t) 2 in two dimensions.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Jamming and tiling in fragmentation of rectangles

We investigate a stochastic process where a rectangle breaks into smaller rectangles through a series of horizontal and vertical fragmentation events. We focus on the case where both the vertical size and the horizontal size of a rectangle are discrete variables. Because of this constraint, the system reaches a jammed state where all rectangles are sticks, that is, rectangles with minimal width. Sticks are frozen as they cannot break any further. The average number of sticks in the jammed state, S , grows as S ≃ A /√2π ln A with rectangle area A in the large-area limit, and remarkably, this behavior is independent of the aspect ratio. The distribution of stick length has a power-law tail, and further, its moments are characterized by a nonlinear spectrum of scaling exponents. Here, we also study an asymmetric breakage process where vertical and horizontal fragmentation events are realized with different probabilities. In this case, there is a phase transition between a weakly asymmetric phase where the length distribution is independent of system size and a strongly asymmetric phase where this distribution depends on system size.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Jamming and Tiling in Aggregation of Rectangles

We study a random aggregation process involving rectangular clusters. In each aggregation event, two rectangles are chosen at random and if they have a compatible side, either vertical or horizontal, they merge along that side to form a larger rectangle. Starting with N identical squares, this elementary event is repeated until the system reaches a jammed state where each rectangle has two unique sides. The average number of frozen rectangles scales as N α in the large-N limit. The growth exponent α = 0.229 ± 0.002 characterizes statistical properties of the jammed state and the time-dependent evolution. We also study an aggregation process where rectangles are embedded in a plane and interact only with nearest neighbors. In the jammed state, neighboring rectangles are incompatible, and these frozen rectangles form a tiling of the two-dimensional domain. Here in this case, the final number of rectangles scales linearly with system size.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

Aggregation-fragmentation-diffusion model for trail dynamics

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. In conclusion, 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.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2016

Escape and finite-size scaling in diffusion-controlled annihilation

In this paper, we study diffusion-controlled single-species annihilation with a finite number of particles. In this reaction-diffusion process, each particle undergoes ordinary diffusion, and when two particles meet, they annihilate. We focus on spatial dimensions d>2 where a finite number of particles typically survive the annihilation process. Using scaling techniques we investigate the average number of surviving particles, M, as a function of the initial number of particles, N. In three dimensions, for instance, we find the scaling law M ~ N 1/3 in the asymptotic regime N»1. We show that two time scales govern the reaction kinetics: the diffusion time scale, T ~ N 2/3 , and the escape time scale, τ ~ N 4/3 . The vast majority of annihilation events occur on the diffusion time scale, while no annihilation events occur beyond the escape time scale.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2016

Kinetics of diffusion-controlled annihilation with sparse initial conditions

Here, we study diffusion-controlled single-species annihilation with sparse initial conditions. In this random process, particles undergo Brownian motion, and when two particles meet, both disappear. We also focus on sparse initial conditions where particles occupy a subspace of dimension δ that is embedded in a larger space of dimension d. Furthermore, we find that the co-dimension Δ = d - δ governs the behavior. All particles disappear when the co-dimension is sufficiently small, Δ ≤ 2; otherwise, a finite fraction of particles indefinitely survive. We establish the asymptotic behavior of the probability S(t) that a test particle survives until time t. When the subspace is a line, δ = 1, we find inverse logarithmic decay, $S\sim {(\mathrm{ln}t)}^{-1}$, in three dimensions, and a modified power-law decay, $S\sim (\mathrm{ln}t){t}^{-1/2}$, in two dimensions. In general, the survival probability decays algebraically when Δ < 2, and there is an inverse logarithmic decay at the critical co-dimension Δ = 2.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2016

Kinetics of Aggregation with Choice

Here we generalize the ordinary aggregation process to allow for choice. In ordinary aggregation, two random clusters merge and form a larger aggregate. In our implementation of choice, a target cluster and two candidate clusters are randomly selected and the target cluster merges with the larger of the two candidate clusters.We study the long-time asymptotic behavior and find that as in ordinary aggregation, the size density adheres to the standard scaling form. However, aggregation with choice exhibits a number of different features. First, the density of the smallest clusters exhibits anomalous scaling. Second, both the small-size and the large-size tails of the density are overpopulated, at the expense of the density of moderate-size clusters. Finally, we also study the complementary case where the smaller candidate cluster participates in the aggregation process and find an abundance of moderate clusters at the expense of small and large clusters. Additionally, we investigate aggregation processes with choice among multiple candidate clusters and a symmetric implementation where the choice is between two pairs of clusters.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2015

Irreversible reactions and diffusive escape: Stationary properties

We study three basic diffusion-controlled reaction processes—annihilation, coalescence, and aggregation. We examine the evolution starting with the most natural inhomogeneous initial configuration where a half-line is uniformly filled by particles, while the complementary half-line is empty. We show that the total number of particles that infiltrate the initially empty half-line is finite and has a stationary distribution. We determine the evolution of the average density from which we derive the average total number N of particles in the initially empty half-line; e.g. for annihilation $\langle N\rangle = \frac{3}{16}+\frac{1}{4\π}$ . For the coalescence process, we devise a procedure that in principle allows one to compute P(N), the probability to find exactly N particles in the initially empty half-line; we complete the calculations in the first non-trivial case (N = 1). As a by-product we derive the distance distribution between the two leading particles.

97 MATHEMATICS AND COMPUTING↗