Bicomponent aggregation in finite systems

The mixing state of a bicomponent population of clusters (granules) is characterized by the normalized variance of excess solute, χ, a parameter that measures the deviation of the composition of each granule from the overall mean. For certain initial conditions and types of kernels, this parameter is constant provided that the population is infinite in size. Here we examine the parameter χ in finite populations and we find that it always decreases. We then apply the analysis to Monte Carlo algorithms that use finite populations to simulate aggregation in infinite systems and find that, while increasing the number of Monte Carlo particles converges the solution to the result for the infinite system, this convergence depends on the degree of homogeneity. In particular, as the degree of homogeneity of non-gelling kernels approaches 1, the number of Monte Carlo particles required to maintain acceptable accuracy with respect to χ in the scaling limit may be prohibitively high.