https://hal-upec-upem.archives-ouvertes.fr/hal-00731537Fournier, NicolasNicolasFournierLAMA - Laboratoire d'Analyse et de Mathématiques Appliquées - UPEM - Université Paris-Est Marne-la-Vallée - Fédération de Recherche Bézout - UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12 - CNRS - Centre National de la Recherche ScientifiquePhilippe, LaurencotLaurencotPhilippeMarcus-Lushnikov processes, Smoluchowski's and Flory's modelsHAL CCSD2009Marcus-Lushnikov processSmoluchowski's coagulation equationFlory's modelgelation[MATH.MATH-PR] Mathematics [math]/Probability [math.PR]Fournier, Nicolas2012-09-13 09:50:272022-01-15 04:01:002012-09-13 09:50:27enJournal articles10.1016/j.spa.2008.02.0031The 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 {\it strongly gelling} kernel behaving as $K(x,y)=x^\alpha y + x y^\alpha$ for some $\alpha\in (0,1]$. In such a case, it is well-known that {\it 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 cut-off 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 cut-off 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.