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Marcus-Lushnikov processes, Smoluchowski’s and Flory’s models. (English) Zbl 1169.60027
The paper studies the connection between a stochastic coalescence model, the Marcus-Lushnikov process, and two deterministic coagulation equations, the Smoluchowski and Flory equations. The Marcus-Lushnikov process is a finite stochastic particle system in which each particle is entirely characterized by its mass. Each pair of particles with masses \(x\) and \(y\) merges into a single particle at a given rate \(K(x,y)\). The paper studies strongly gelling kernels behaving as \(K(x,y) = x^\alpha y + x y^\alpha\) for some \(\alpha \in (0,1]\). For such kernels it is well-known that gelation occurs, i.e. giant particles emerge. Two possible models for the hydrodynamic limit as the number of particles goes to infinity arise: the Smoluchowski equation, in which the giant particles are inert, and the Flory equation, in which the giant particles interact with the finite ones. The paper shows these limits in the Skorokhod topology using a suitable cut-off coagulation kernel in the Marcus-Lushnikov process. Furthermore, the asymptotic behaviour of the largest particle in the Marcus-Lushnikov process without cut-off is studied, and it is shown that there is only one giant particle. This single particle represents, asymptotically, the lost mass of the solution to the Flory equation. The paper is concluded with a small simulation study, where the theoretical results are demonstrated.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
45K05 Integro-partial differential equations
60G57 Random measures
60H30 Applications of stochastic analysis (to PDEs, etc.)
60F99 Limit theorems in probability theory
Full Text: DOI arXiv
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