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This papers addresses the connection between two classical models of phasetransition phenomena describing different stages of the growth of clusters. TheBecker-D\"oring model (BD) describes discrete-sized clusters through aninfinite 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 $\varepsilon>0$, whichaccounts for the grid size of the state space in the BD model, and recover theLS model in the limit $\varepsilon\to 0$. The connection has been alreadyproven in the context of outgoing characteristic at the boundary $x=0$ for theLS model, when small clusters tend to shrink. The main novelty of this workresides in a new estimate on the growth of small clusters, which behave at afast time scale. Through a rigorous quasi steady state approximation, we deriveboundary conditions for the incoming characteristic case, when small clusterstend to grow.

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