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 rho, determined by the initial data, acts as an order parameter, in which the system shows a phase transition. There is a critical value rho(c) is an element of (0, infinity] characterized by the rate kernel. For rho <= rho(c), there exists a unique equilibrium state omega(rho) and the solution converges strongly to omega(rho). If rho > rho(c), the solution converges only weakly to omega(rho c). In particular, the excess rho - rho(c) gets lost due to the formation of larger and larger clusters. In this regard, the model behaves similarly to the Becker-Doring 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.