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.
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Fournier, N., Giet, JS. Convergence of the Marcus–Lushnikov Process. Methodology and Computing in Applied Probability 6, 219–231 (2004). https://doi.org/10.1023/B:MCAP.0000017714.56667.67
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DOI: https://doi.org/10.1023/B:MCAP.0000017714.56667.67