Marcus–Lushnikov processes, Smoluchowski’s and Flory’s models

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 + x y 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. w 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. o 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.