Phases of a conserved mass model of aggregation with fragmentation at fixed sites

Abstract

To study the effect of quenched disorder in a class of reaction-diffusion systems, we introduce a conserved mass model of diffusion and aggregation in which fragmentation occurs only at certain fixed sites. On most sites, the mass moves as a whole to a nearest neighbor while it leaves the fixed sites only as a single monomer (i.e., chips off). Once the mass leaves any site, it coalesces with the mass present on its neighbor. We study in detail the effect of a single chipping site on the steady state in arbitrary dimensions, with and without bias. In the thermodynamic limit, the system can exist in one of the following three phases. (a) Pinned aggregate (PA) phase in which an infinite aggregate (with mass proportional to the volume of the system) appears at the chipping site with probability one but not in the bulk. (b) Unpinned aggregate (UA) phase in which the infinite aggregate occurs at the chipping site with a probability strictly less than one and can coexist with infinite aggregates in the bulk. (c) Nonaggregate (NA) phase in which there is no infinite cluster. The steady state of the system depends on the dimension and drive. A sitewise inhomogeneous mean field theory predicts that the system exists in the UA phase in all cases. Monte Carlo simulations in one and two dimensions support this prediction in all but one-dimensional, biased case. In the latter case, there is a phase transition from the NA phase to the PA phase as the density is increased. We identify the critical point exactly and calculate the mass distribution in the PA phase. The NA phase and the critical point are studied by Monte Carlo simulations and using scaling arguments. A variant of the above aggregation model is also considered in which total particle number is conserved and chipping occurs at a fixed site, but the particles do not interact with each other at other sites. This model is solved exactly by mapping it to a zero range process. With increasing density, it exhibits a phase transition from the NA phase to the PA phase in all dimensions, irrespective of bias. The free-particle model is also solved with an extensive number of chipping sites with random chipping rates and we argue that it qualitatively describes the behavior of the aggregation model with extensive disorder.