Spatial fluctuations in reaction-diffusion systems: A model for exponential growth

The spatial fluctuations in an exactly soluble model for the irreversible aggregation of clusters are treated. The model is characterized byrate constants K_ij=i+j for the clustering of ani- and aj-mer, anddiffusion constants D_j=D. It is assumed thatD1 (reaction-limited aggregation). Explicit expressions for the correlation functions at equal and at different times are calculated. The equal-time correlation functions show scaling behavior in the scaling limit. The correlation length remains finite ast, and the fluctuations becomelarge at large times (tt_D) inany dimension. The crossover timet_D, at which the mean field theory (Smoluchowski's equation) breaks down, is given byt_DInD. These exact results imply that the upper critical dimension of this model isd_c= and, hence, that there isno unique upper critical dimension in models for the irreversible aggregation of clusters.