The exchange-driven growth model: basic properties and longtime behavior

The exchange-driven growth model describes a process in which pairs of clusters interact through the exchange of single monomers. The rate of exchange is given by an interaction kernel $K$ which depends on the size of the two interacting clusters. Well-posedness of the model is established for kernels growing at most linearly and arbitrary initial data. The longtime behavior is established under a detailed balance condition on the kernel. The total mass density $\varrho$, determined by the initial data, acts as an order parameter, in which the system shows a phase transition. There is a critical value $\varrho_c\in (0,\infty]$ characterized by the rate kernel. For $\varrho \leq \varrho_c$, there exists a unique equilibrium state $\omega^\varrho$ and the solution converges strongly to $\omega^\varrho$. If $\varrho > \varrho_c$ the solution converges only weakly to $\omega^{\varrho_c}$. In particular, the excess $\varrho - \varrho_c$ gets lost due to the formation of larger and larger clusters. In this regard, the model behaves similarly to the Becker-D\"oring equation. The main ingredient for the longtime behavior is the free energy acting as Lyapunov function for the evolution. It is also the driving functional for a gradient flow structure of the system under the detailed balance condition.