193 0022-4715/01/0700-0193$19.50/0 © 2001 Plenum Publishing Corporation Journal of Statistical Physics, Vol. 104, Nos. 1/2, 2001 The Discrete Coagulation Equations with Collisional Breakage Philippe Laurençot 1 and Dariusz Wrzosek 2 1 CNRS & Institut Elie Cartan - Nancy, Université de Nancy I, BP 239, F-54506 Vandœuvre- le `s-Nancy cedex, France; e-mail: laurenco@iecn.u-nancy.fr 2 Institute of Applied Mathematics and Mechanics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland; e-mail: darekw@mimuw.edu.pl Received July 18, 2000; revised January 26, 2001 The discrete coagulation equations with collisional breakage describe the dynamics of cluster growth when clusters undergo binary collisions resulting either in coalescence or breakup with possible transfer of matter. Each of these two events may happen with an a priori prescribed probability depending for instance on the sizes of the colliding clusters. We study the existence, density conservation and uniqueness of solutions. We also consider the large time behaviour and discuss the possibility of the occurrence of gelation in some par- ticular cases. KEY WORDS: Cluster growth; coalescence; collisional breakage; existence of solutions; propagation of moments. 1. INTRODUCTION Coagulation-fragmentation processes naturally occur in the dynamics of cluster growth and describe the way a system of clusters can merge to form larger ones or fragment to form smaller ones. Models of cluster growth arise in a wide variety of situations, including aerosol science, astrophysics, colloidal chemistry, polymer science, and biology. In the model considered in this paper the clusters are assumed to be discrete, that is, they consist of a finite number of identical elementary particles. The basic reactions between clusters taken into account are the coalescence of two clusters to form a larger one and the breakage of clusters into smaller pieces. At least two physical mechanisms have been considered to describe the latter