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 ${\varrho }_c\in (0,\mathrm{\infty }]$ characterized by the rate kernel. For $\varrho \le {\varrho }_c$, 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.