Scaling Theory for Migration-Driven Aggregate Growth

We give a comprehensive description for the irreversible growth of aggregates by migration from small to large aggregates. For a homogeneous rate $K(i;j)$ at which monomers migrate from aggregates of size $i$ to those of size $j$, that is, $K(ai;aj)\sim a^{\lambda }K(i;j)$, the mean aggregate size grows with time as $t^{1/(2-\lambda )}$ for $\lambda <2\mathrm{}$. The aggregate size distribution exhibits distinct regimes of behavior that are controlled by the scaling properties of the migration rate from the smallest to the largest aggregates. Our theory applies to diverse phenomena such as the distribution of city populations, late stage coarsening of nonsymmetric binary systems, and models for wealth exchange.