Large-time behavior of the Smoluchowski equations of coagulation

Abstract

Large-time asymptotic solutions of the Smoluchowski equations for rapid coagulation are displayed for two common forms of the reaction kernel, namely, $R_{\mathrm{ij}}=i^{\omega }{j}^{\omega }$ and $R_{\mathrm{ij}}=i^{\omega }+j^{\omega }$. If no singularity occurs at finite time, it is shown that indices similar to $\tau$ and $\sigma$ in percolation can be defined: The concentrations approach a power-law behavior with an exponential correction the importance of which diminishes as a given power of time.