The articles in this special issue of Communications in Mathematical Sciences were written for the first meeting of the GdR GRIP (Research Group on Particle Interactions) held in Paris in May 2003

We present here an algorithm to simulate the motion of rigid bodies subject to a non-overlapping constraint, and which tend to aggregate when they get close to each other. The motion is induced by external forces. Two types of forces are considered here: drift force induced by the action of a surrounding fluid whose motion is prescribed, and stochastic forces modelling random shocks of molecules on the surface of the bodies. The numerical approach fits into the general framework of granular flow modelling.

The present paper is devoted to the kinetic modeling of coalescence in turbulent gas-droplet flows. A new approach is proposed for the calculation of the collision probability, that takes into account the correlations induced by the effect of the gas on the droplet motion. The key ingredient is to replace the simple distribution function f⁽¹⁾_p (t, x, v, r), which is classically used for the description of a spray at the kinetic level, by the joint distribution function, f⁽¹⁾ _pg (t, x, v, u, r), which explicitely depends on the fluctuating gas velocity u at the droplet position.

This paper is meant as an introduction to some of the most classical models in the theory of fragmentation-coagulation. The main models presented are the Becker-Döring, fragmentation-coagulation (discrete or continuous) and Lifshitz-Slyozov ones. Rather than focusing on mathematical technicalities, we have chosen to insist on the physical ideas behind their derivation, in order to present them in a unified framework. The unifying physical principle in this context is the mass action principle, which we expose in detail, our philosophy being that these models may be thought of as technical variations on this theme. We then present some qualitative properties of the models, which include saturation, criticality, and dissipation. The second part of the paper collects some mathematical tools which are of recurrent use in this context, namely the use of moments, of the Laplace transform, and of Lyapunov functions.

Consider a spatially homogeneous infinite particle system in which coalescence and elastic collisions occur. The Boltzmann-Smoluchowski equation describes the evolution of the concentration f(t, m, v) of particles of mass m and velocity v at time t ≥ 0. Using a stochastic version of this equation, we give an exact simulation scheme and we study the asymptotics of solutions for large times.

For fragmentations in which particles split even faster when their mass is smaller, it is possible to observe a decrease of the total mass of the system, due to the reduction into dust. We investigate here this appearance of dust for a large class of deterministic and random fragmentations.

In this paper, supposing that either the initial data is small or the fragmentation phenomenon dominates the coagulation, we associate a nonlinear stochastic process with any solution of the mass-flow equation obtained from the discrete Smoluchowski coagulation fragmentation equation by a natural change of variables. This enables us to deduce uniqueness for the mass flow equation and therefore for the corresponding Smoluchowski equation thanks to a coupling argument.

We review the derivation of macroscopic limits of the Becker-Döring equations. We show that those limits have the structure of a gradient flow even though the Becker-Döring equations themselves do not allow for such an interpretation.

We consider a stochastic particle model for coagulating particles, whose free motion is Brownian, with diffusivity given by Einstein's law. We present in outline a derivation from this model of a spatially inhomogeneous version of Smoluchowski's coagulation equation. Some analytic results on existence, uniqueness and mass conservation for the limit equation are also presented.