Longtime behavior for exchange-driven growth models

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 which depends on the size of the two interacting clusters. The model was recently obtained as the mean-field limit of stochastic particle systems (zero-range process).

Well-posedness of the model is discussed under suitable growth bounds on the kernel and arbitrary initial data. The central part of the talk considers the longtime behavior under a detailed balance condition on the kernel. The total mass density, determined by the initial data, acts as an order parameter, in which the system shows a phase transition in the following sense: There is a critical mass density characterized by the rate kernel. In the subcritical regime, there exists a unique equilibrium state, and the solution converges in a strong sense (conservation of mass). In the supercritical regime, the solution converges only in a weak sense, where excess mass gets lost due to the formation of larger and larger clusters.

In this regard, the model behaves similarly to the Becker-Döring 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 assumption of detailed balance.

The talk closes with an outlook on work in progress together with Bob Pego and Juan Velazquez on related models without detailed balance showing stable oscillations as the longtime behavior.