Stationary Kolmogorov Solutions of the Smoluchowski Aggregation Equation with a Source Term

Abstract

In this paper we show how the method of Zakharov transformations may be used to analyze the stationary solutions of the Smoluchowski aggregation equation for arbitrary homogeneous kernel. The resulting massdistributions are of Kolmogorov type in the sense that they carry a constant flux of mass from small masses to large. We derive a ``locality criterion'', expressed in terms of the asymptotic properties of the kernel, that must be satisfied in order for the Kolmogorov spectrum to be an admissiblesolution. Whether a given kernel leads to a gelation transition or not can be determined by computing the mass capacity of the Kolmogorov spectrum. As an example, we compute the exact stationary state for the family of kernels,$K_ζ(m_1,m_2)=(m_1m_2)^{ζ/2}$ which includes both gelling and non-gelling cases, reproducing the known solution in the case $ζ=0$. Surprisingly, the Kolmogorov constant is the same for all kernels in this family.

This article is an expanded version of a talk given at IHP workshop "Dynamics, Growth and Singularities of Continuous Media", Paris July 2003. Updated 01/04/04. Revised version with additional discussion, references added, several typographical errors corrected. Revised version accepted for publication by Phys. Rev. E