## Preprint 2014-35

## Julien Deschamps, Erwan Hingant, Romain Yvinec:

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

### Abstract:

We investigate the connection between two classical models of phase transition phenomena, the (discrete size) stochastic Becker-D¨oring equations and the (continuous size) deterministic Lifshitz-Slyozov equation. For general coefficients and initial data, we introduce a scaling parameter and show that the empirical measure associated to the stochastic Becker- D¨oring system converges in law to the weak solution of the Lifshitz-Slyozov equation when the parameter goes to 0. Contrary to previous studies, we use a weak topology that includes the boundary of the state space allowing us to rigorously derive a boundary value for the Lifshitz-Slyozov model in the case of incoming characteristics. It is the main novelty of this work and it answers to a question that has been conjectured or suggested by both mathematicians and physicists. We emphasize that the boundary value depends on a particular scaling (as opposed to a modeling choice) and is the result of a separation of time scale and an averaging of fast (fluctuating) variables