Particle Survival and Polydispersity in Aggregation

– We study the probability, PS(t), of a cluster to remain intact in one-dimensional cluster-cluster aggregation when the cluster diffusion coefficient scales with size as D(s) ∼ s . PS(t) exhibits a stretched exponential decay for γ < 0 and the power-laws t −3/2 for γ = 0, and t for 0 < γ < 2. A random walk picture explains the discontinuous and non-monotonic behavior of the exponent. The decay of PS(t) determines the polydispersity exponent, τ , which describes the size distribution for small clusters. Surprisingly, τ (γ) is a constant τ = 0 for 0 < γ < 2. Many models of aggregation phenomena lead to scale-invariance: the average cluster size increases as a power-law, S(t) ∼ t, which defines a dynamical exponent z. This kind of behavior is met in various contexts ranging from chemical engineering to materials science to atmosphere research to, ultimately, even astrophysics [1]. It is of interest to explore the statistics of aggregation as a dynamical process, beyond the lengthand timescales defined through z. In this Letter we introduce a new quantity in aggregation systems, the cluster survival, defined as the probability PS(t), that a cluster present at t = 0 remains unaggregated until time t. This is a first passage problem [2] in a many body system and analogous to persistence which is often studied by measuring the fraction of a system that preserves its initial condition for all times [0, t] [3]. The cluster survival turns out to decay in a nontrivial and counterintuitive manner. The behavior can be understood by a mean-field like random walk analysis. However, even on the mean-field level the question reduces to a novel, unsolved random walk problem, which we analyse in the long time limit. More importantly, by solving the decay of the cluster survival we are able to determine the polydispersity exponent characterising the cluster size distribution. We concentrate on a common and important example: diffusion–limited cluster–cluster aggregation (DLCA) [4]. In the lattice version of DLCA any set of nearest neighbor occupied lattice sites is identified as a cluster. Each of these performs a random walk with a size dependent diffusion constant, D(s) ∼ s , where γ is the diffusion exponent. Colliding clusters are merged together and the aggregate diffuses either faster (γ > 0) or slower than before (γ < 0). In the following cluster survival is investigated in the one-dimensional case for