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Article Dans Une Revue Stochastic Processes and their Applications Année : 2009

Marcus-Lushnikov processes, Smoluchowski's and Flory's models

Résumé

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). We consider a strongly gelling kernel behaving as K(x, y) = x α y + xy α for some α ∈ (0, 1]. In such a case, it is well-known that gelation occurs, that is, giant particles emerge. Then two possible models for hydrodynamic limits of the Marcus-Lushnikov process arise: the Smoluchowski equation, in which the giant particles are inert, and the Flory equation, in which the giant particles interact with finite ones. We show that, when using a suitable cutoff coagulation kernel in the Marcus-Lushnikov process and letting the number of particles increase to infinity, the possible limits solve either the Smoluchowski equation or the Flory equation. We also study the asymptotic behaviour of the largest particle in the Marcus-Lushnikov process without cutoff and show that there is only one giant particle. This single giant particle represents, asymptotically, the lost mass of the solution to the Flory equation.
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Dates et versions

hal-00627096 , version 1 (28-02-2018)

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Nicolas Fournier,, Philippe Laurençot. Marcus-Lushnikov processes, Smoluchowski's and Flory's models. Stochastic Processes and their Applications, 2009, 119 (1), pp.167-189. ⟨10.1016/j.spa.2008.02.003⟩. ⟨hal-00627096⟩
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