Asymptotic behavior of the generalized Becker-D\"oring equations for general initial data

We prove the following asymptotic behavior for solutions to the generalized Becker-D\"oring system for general initial data: under a detailed balance assumption and in situations where density is conserved in time, there is a critical density $\rho_s$ such that solutions with an initial density $\rho_0 \leq \rho_s$ converge strongly to the equilibrium with density $\rho_0$, and solutions with initial density $\rho_0 > \rho_s$ converge (in a weak sense) to the equilibrium with density $\rho_s$. This extends the previous knowledge that this behavior happens under more restrictive conditions on the initial data. The main tool is a new estimate on the tail of solutions with density below the critical density.