Statistics of aggregates

Aggregation phenomena of elementary particles into clusters has received co nsiderable attention during the past few decades. We adopt here a stochasti c approach for the modeling of these phenomena. More precisely, we formulat e the problem in the following dynamical setup: given a population of n dis cernible atoms partitioned into p discernible (model 1) or indiscernible (m odel 2) groups, how does a new atom eventually connect to any of these p gr oups forming up a new partition of n + 1 atoms into a certain amount of clu sters? Nucleation is said to occur when the inserted atom does not connect tit nucleates), whereas aggregation takes place if it does (clusters coales ce). Depending on this local "logic" of pattern formation, the asymptotic s tructure of groups can be quite different, in the thermodynamic limit n --> infinity. These studies are the main purpose of this work. Understanding t hese aggregation phenomena requires first to derive the fragment size distr ibutions (that is, the number P of fragments, or clusters, and the number N -m of size-m fragments with m constitutive atoms), as a function of the con trol 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 t hermodynamic limit n --> infinity. It is shown, making extensive use of com binatorics, that two regimes are to be distinguished: the one of weakly con nected aggregates where nucleation dominates aggregation and the one of str ongly connected aggregates where aggregation dominates nucleation. In the f irst ("diluted") regime, the number of clusters P(n) always diverges as n - -> infinity, 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 ("cond ensed") regime, the number of groups P(n) and size-m groups N-m(n) may conv erge in the thermodynamic limit, with a special role played by the geometri c and Poisson distributions. The asymptotic variables become observable mac roscopically. This work is therefore aimed toward a better understanding of the fundamentals involved in clusters' formation processes.