Convergence of the Marcus–Lushnikov Process

Abstract

The Smoluchowski coagulation equation describes the concentration c(t,x) of particles of mass x ∈ [ 0,∞] at the instant t ≥ 0, in an infinite system of coalescing particles. It is well-known that in some cases, gelation occurs: a particle with infinite mass appears. But this infinite particle is inert, in the sense that it does not interact with finite particles. We consider the so-called Marcus–Lushnikov process, which is a stochastic finite system of coalescing particles. This process is expected to converge, as the number of particles tends to infinity, to a solution of the Smoluchowski coagulation equation. We show that it actually converges, for t ∈ [0,∞], to a modified Smoluchowski equation, which takes into account a possible interaction between finite and infinite particles.