9 - Stochastic models of aggregation with injection

Summary

A generalized aggregation model of charged particles that diffuse and coalesce randomly in discrete space-time is studied, numerically and analytically. A statistically invariant steady state is established when randomly charged particles are uniformly and continuously injected. The exact steadystate size distribution obeys a power law whose exponent depends on the type of injection. The stability of the power-law size distribution is proved. The spatial correlations of the system are analyzed by a powerful new method, the interval distribution of a level set, and a scaling relation is obtained.

Introduction

The study of far-from-equilibrium systems has attracted much attention in the last two decades. Though many macroscopic phenomena in nature, such as turbulence, lightning, earthquakes, fracture, erosion, the formation of clouds, aerosols, and interstellar dusts, are typical far-from-equilibrium problems, no unified view has yet been established. The substantial difficulties in studying such systems are the following. First, far-from-equilibrium systems satisfy neither detailed balance nor, at the macroscopic level, the equipartition principle. Second, the system is usually open to an outside source. A common method to describe such systems is by abstracting the macroscopic essential features of the observed system and constructing a model in macroscopic terms irrespective of the microscopic (molecular) dynamics. In other words, we make a far-from-equilibrium model by assuming appropriate irreversible rules for the macroscopic dynamics.