Quasi steady state approximation of the small clusters in Becker-D\"oring equations leads to boundary conditions in the Lifshitz-Slyozov limit

This papers addresses the connection between two classical models of phase transition phenomena describing different stages of the growth of clusters. The Becker-D\"oring model (BD) describes discrete-sized clusters through an infinite set of ordinary differential equations. The Lifshitz-Slyozov equation (LS) is a transport partial differential equation on the continuous half-line $x\in (0,+\infty)$. We introduce a scaling parameter $\varepsilon>0$, which accounts for the grid size of the state space in the BD model, and recover the LS model in the limit $\varepsilon\to 0$. The connection has been already proven in the context of outgoing characteristic at the boundary $x=0$ for the LS model, when small clusters tend to shrink. The main novelty of this work resides in a new estimate on the growth of small clusters, which behave at a fast time scale. Through a rigorous quasi steady state approximation, we derive boundary conditions for the incoming characteristic case, when small clusters tend to grow.