Mass-transport models with fragmentation and aggregation

There has been intense research interest in phase transitions in mass-transport and growth models involving adsorption and desorption, fragmentation, diffusion, and aggregation. These processes are ubiquitous in nature and arise in a large number of seemingly diverse systems such as growing interfaces [1,2], colloidal suspensions [3], polymer gels [4], river networks [5], granular materials [6], traffic flows [7], etc. In these systems, different nonequilibrium states arise if the rates of the underlying microscopic processes are varied. Conservation laws also play an important role in determining both the time-dependent behavior and steady states of such systems. As these steady states are usually not described by the Gibbs distribution, they are hard to determine. However, much insight on this issue has been gained by studying lattice models. Due to their simplistic nature, these models can be treated either exactly or via a mean-field (MF) approach [8–13]. They are also simple to implement numerically using Monte Carlo (MC) techniques [14,15]. In this article, we present a pedagogical discussion of the modeling and simulation of mass-transport and growth phenomena. We discuss analytical and numerical techniques in the context of mass-transport models where the elementary move is the fragmentation of mass k, and its subsequent diffusion to a neighboring site where it aggregates. The k-chip models that we study here are interesting in physical situations where the deposited material consists of polymers. We study the MF limit of these models, focusing on the steady-state mass distribution (P(m) vs. m), which is characterized by k branches. We also compare the MF results with MC simulations in d1⁄4 1, 2. This article is organized as follows. In Section 2, we present a framework for mass-transport models in terms of the rate (evolution) equations for P(m, t), the