From a stochastic Becker-Döring model to the Lifschitz-Slyozov equation with boundary value

We deal with the convergence in law of the stochastic Becker-Döring process to the Lifschitz-Slyozov partial differential equation, up to a small scaling parameter. The former is a probabilistic model for the lengthening/shrinking dynamics of a finite number and discrete size clusters, while the latter is seen as its infinite number and continuous size extension. In the Becker-Döring model, the clusters are assumed to increase or decrease their size (number of particles in a cluster) by addition or subtraction of only one single particle at a time (stepwise coagulation and fragmentation) without regarding the space structure. More precisely, in this model, the transitions are assumed to be Markovian and actually related to some random Poisson point measures. The lengthening rates depend on the size, the number of clusters of this size and the number of free particles throught a Law of Mass Action. The fragmentation rates depend on the size and the number of clusters of this size, through a spontaneous shricking (exponential law). The evolution of the configuration of the system is then described thanks to its empirical measure. It starts with a finite number of clusters and particles. So that, the state space of the model is finite (but possibly large) and bounded by the number of particles and clusters of all possible sizes up to the maximal one (given by the total number of particles in the system). Under an appropriate scaling of the rates parameters, the number of monomers and the sizes of clusters, we construct a rescaled measure-valued stochastic process from the empirical measure of the Becker-Döring model. We prove the convergence in law of this process towards a measure solution of the Lifschitz-Slyozov equation. This equation is of transport type with a nonlinear flux coupling the particle variable. The necessity of prescribing a boundary value at the minimal size naturally appears in the case of incoming characteristics. The value of the latter is still an open-debated question for this continuous model. The probabilistic approach of this work allows us to rigorously derive a boundary value as a result of a particular scaling (as opposed to a modeling choice) of the original discrete model. The proof of this result is mainly based on an adiabatic procedure, the boundary condition being the result of a separation of time scale and an averaging of a fast (fluctuating) variable.