Particle Survival and Polydispersity in Aggregation

We study the probability, $P_S(t)$, of a cluster to remain intact in one-dimensional cluster-cluster aggregation when the cluster diffusion coefficient scales with size as $D(s)\sim s^{\gamma }$. $P_S(t)$ exhibits a stretched exponential decay for $\gamma <0$ and the power-laws $t^{-3\mathrm{/}2}$ for $\gamma =0$, and $t^{-2\mathrm{/}(2-\gamma )}$ for $0<\gamma <2$. A random walk picture explains the discontinuous and non-monotonic behavior of the exponent. The decay of $P_S(t)$ determines the polydispersity exponent, $\tau$, which describes the size distribution for small clusters. Surprisingly, $\tau (\gamma )$ is a constant $\tau =0$ for $0<\gamma <2$.