Spatial fluctuations in reaction-limited aggregation

Abstract

A general method is used for describing reaction-diffusion systems, namely van Kampen's “method of compounding moments,” to study the spatial fluctuations in reaction-limited aggregation processes. The general formalism used here and in subsequent publications is developed. Then a particular model is considered that is of special interest, since it describes the occurrence of a phase transition (gelation). The corresponding rate constants for the reaction between two clusters of sizei and sizej areKij=ij (i, j=1, 2,⋯). For thediffusion constants Dj of clusters of sizej the following class of models is considered:Dj=D if 1⩽J⩽s andDj=0 ifj>s. The casess=∞ ands<∞ are studied separately. For the models=∞ the equal-time and the two-time correlation functions are calculated; this modelbreaks down at the gel point. The breakdown is characterized by a divergence of the density fluctuations, and is caused by the large mobility of large clusters. For all models withs<∞ the density fluctuations remain finite attc, and the equal-time correlation functions in the pre- and in the post-gel stage are calculated. Many explicit and asymptotic results are given. From the exact solution the upper critical dimension in this gelling model isdc=2.