Statistics of aggregates

Abstract

Aggregation phenomena of elementary particles into clusters has received considerable attention during the past few decades. We adopt here a stochastic approach for the modeling of these phenomena. More precisely, we formulate the problem in the following dynamical setup: given a population of n discernible atoms partitioned into p discernible (model 1) or indiscernible (model 2) groups, how does a new atom eventually connect to any of these p groups forming up a new partition of n+1 atoms into a certain amount of clusters? Nucleation is said to occur when the inserted atom does not connect (it nucleates), whereas aggregation takes place if it does (clusters coalesce). Depending on this local “logic” of pattern formation, the asymptotic structure of groups can be quite different, in the thermodynamic limit N→∞. These studies are the main purpose of this work. Understanding these aggregation phenomena requires first to derive the fragment size distributions (that is, the number P of fragments, or clusters, and the number n_m of size-m fragments with m fragments with constitutive atoms), as a function of the control parameter which is chosen here to be the average number of atoms 〈N〉. As 〈N〉 approaches infinity, we derive the study of these variables in the thermodynamic limit n → ∞. It is shown, making extensive use of combinatorics, that two regimes are to be distinguished: the one of weakly connected aggregates where nucleation dominates aggregation and the one of strongly connected aggregates where aggregation dominates nucleation. In the first (“diluted”) regime, the number of clusters P(n) always diverges as n → ∞, the asymptotic equivalent of which being under control in most cases. Large deviation results are shown to be available. Concerning N _m(n), distinct behaviours are observed in models 1 and 2. In the second (“condensed”) regime, the number of groups P(n) and size-m groups N _m(n) may converge in the thermodynamic limit, with a special role played by the geometric and Poisson distributions. The asymptotic variables become observable macroscopically. This work is therefore aimed toward a better understanding of the fundamentals involved in clusters' formation processes.