Stochastic Aggregation: Scaling Properties

We study scaling properties of stochastic aggregation processes in one dimension. Numerical simulations for both diffusive and ballistic transport show that the mass distribution is characterized by two independent nontrivial exponents corresponding to the survival probability of particles and monomers. The overall behavior agrees qualitatively with the mean-field theory. This theory also provides a useful approximation for the decay exponents, as well as the limiting mass distribution.Comment: 6 pages, 7 figure

Ballistic Annihilation Kinetics: The Case of Discrete Velocity Distributions

The kinetics of the annihilation process, $A+A\to 0$, with ballistic particle motion is investigated when the distribution of particle velocities is {\it discrete}. This discreteness is the source of many intriguing phenomena. In the mean field limit, the densities of different velocity species decay in time with different power law rates for many initial conditions. For a one-dimensional symmetric system containing particles with velocity 0 and $\pm 1$, there is a particular initial state for which the concentrations of all three species as decay as $t^{-2/3}$. For the case of a fast ``impurity'' in a symmetric background of $+$ and $-$ particles, the impurity survival probability decays as $\exp(-{\rm const.}\times \ln^2t)$. In a symmetric 4-velocity system in which there are particles with velocities $\pm v_1$ and $\pm v_2$, there again is a special initial condition where the two species decay at the same rate, t^{-\a}, with \a\cong 0.72. Efficient algorithms are introduced to perform the large-scale simulations necessary to observe these unusual phenomena clearly.Comment: 18 text pages, macro file included, hardcopy of 9 figures available by email request to S

Random Geometric Series

Integer sequences where each element is determined by a previous randomly chosen element are investigated analytically. In particular, the random geometric series x_n=2x_p with 0<=p<=n-1 is studied. At large n, the moments grow algebraically, n^beta(s) with beta(s)=2^s-1, while the typical behavior is x_n n^ln 2. The probability distribution is obtained explicitly in terms of the Stirling numbers of the first kind and it approaches a log-normal distribution asymptotically.Comment: 6 pages, 2 figure

Life and Death at the Edge of a Windy Cliff

The survival probability of a particle diffusing in the two dimensional domain $x>0$ near a ``windy cliff'' at $x=0$ is investigated. The particle dies upon reaching the edge of the cliff. In addition to diffusion, the particle is influenced by a steady ``wind shear'' with velocity $\vec v(x,y)=v\,{\rm sign}(y)\,\hat x$, \ie, no average bias either toward or away from the cliff. For this semi-infinite system, the particle survival probability decays with time as $t^{-1/4}$, compared to $t^{-1/2}$ in the absence of wind. Scaling descriptions are developed to elucidate this behavior, as well as the survival probability within a semi-infinite strip of finite width $|y|<w$ with particle absorption at $x=0$. The behavior in the strip geometry can be described in terms of Taylor diffusion, an approach which accounts for the crossover to the $t^{-1/4}$ decay when the width of the strip diverges. Supporting numerical simulations of our analytical results are presented.Comment: 13 pages, plain TeX, 5 figures available upon request to SR (submitted to J. Stat. Phys.

Dynamics of Multi-Player Games

We analyze the dynamics of competitions with a large number of players. In our model, n players compete against each other and the winner is decided based on the standings: in each competition, the mth ranked player wins. We solve for the long time limit of the distribution of the number of wins for all n and m and find three different scenarios. When the best player wins, the standings are most competitive as there is one-tier with a clear differentiation between strong and weak players. When an intermediate player wins, the standings are two-tier with equally-strong players in the top tier and clearly-separated players in the lower tier. When the worst player wins, the standings are least competitive as there is one tier in which all of the players are equal. This behavior is understood via scaling analysis of the nonlinear evolution equations.Comment: 8 pages, 8 figure

Kinetics of Heterogeneous Single-Species Annihilation

We investigate the kinetics of diffusion-controlled heterogeneous single-species annihilation, where the diffusivity of each particle may be different. The concentration of the species with the smallest diffusion coefficient has the same time dependence as in homogeneous single-species annihilation, A+A-->0. However, the concentrations of more mobile species decay as power laws in time, but with non-universal exponents that depend on the ratios of the corresponding diffusivities to that of the least mobile species. We determine these exponents both in a mean-field approximation, which should be valid for spatial dimension d>2, and in a phenomenological Smoluchowski theory which is applicable in d<2. Our theoretical predictions compare well with both Monte Carlo simulations and with time series expansions.Comment: TeX, 18 page

Stable Distributions in Stochastic Fragmentation

We investigate a class of stochastic fragmentation processes involving stable and unstable fragments. We solve analytically for the fragment length density and find that a generic algebraic divergence characterizes its small-size tail. Furthermore, the entire range of acceptable values of decay exponent consistent with the length conservation can be realized. We show that the stochastic fragmentation process is non-self-averaging as moments exhibit significant sample-to-sample fluctuations. Additionally, we find that the distributions of the moments and of extremal characteristics possess an infinite set of progressively weaker singularities.Comment: 11 pages, 5 figure

Two-Scale Annihilation

The kinetics of single-species annihilation, $A+A\to 0$, is investigated in which each particle has a fixed velocity which may be either $\pm v$ with equal probability, and a finite diffusivity. In one dimension, the interplay between convection and diffusion leads to a decay of the density which is proportional to $t^{-3/4}$. At long times, the reactants organize into domains of right- and left-moving particles, with the typical distance between particles in a single domain growing as $t^{3/4}$, and the distance between domains growing as $t$. The probability that an arbitrary particle reacts with its $n^{\rm th}$ neighbor is found to decay as $n^{-5/2}$ for same-velocity pairs and as $n^{-7/4}$ for $+-$ pairs. These kinetic and spatial exponents and their interrelations are obtained by scaling arguments. Our predictions are in excellent agreement with numerical simulations.Comment: revtex, 5 pages, 5 figures, also available from http://arnold.uchicago.edu/~eb

Kinetics of Clustering in Traffic Flows

We study a simple aggregation model that mimics the clustering of traffic on a one-lane roadway. In this model, each ``car'' moves ballistically at its initial velocity until it overtakes the preceding car or cluster. After this encounter, the incident car assumes the velocity of the cluster which it has just joined. The properties of the initial distribution of velocities in the small velocity limit control the long-time properties of the aggregation process. For an initial velocity distribution with a power-law tail at small velocities, \pvim as $v \to 0$, a simple scaling argument shows that the average cluster size grows as n \sim t^{\va} and that the average velocity decays as v \sim t^{-\vb} as $t\to \infty$. We derive an analytical solution for the survival probability of a single car and an asymptotically exact expression for the joint mass-velocity distribution function. We also consider the properties of spatially heterogeneous traffic and the kinetics of traffic clustering in the presence of an input of cars.Comment: 18 pages, Plain TeX, 2 postscript figure

Coarsening in a Driven Ising Chain with Conserved Dynamics

We study the low-temperature coarsening of an Ising chain subject to spin-exchange dynamics and a small driving force. This dynamical system reduces to a domain diffusion process, in which entire domains undergo nearest-neighbor hopping, except for the shortest domains -- dimers -- which undergo long-range hopping. This system is characterized by two independent length scales: the average domain length L(t)~t^{1/2} and the average dimer hopping distance l(t)~ t^{1/4}. As a consequence of these two scales, the density C_k(t) of domains of length k does not obey scaling. This breakdown of scaling also leads to the density of short domains decaying as t^{-5/4}, instead of the t^{-3/2} decay that would arise from pure domain diffusion.Comment: 7 pages, 9 figures, revtex 2-column forma