|TITLE||The exchange-driven growth model: Basic properties and longtime behaviour|
|ABSTRACT||The exchange-driven growth model describes a process in which pairs
of clusters interact and exchange a single monomer. 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 ρ, determined by the initial data, acts as an order parameter, in which the system shows a phase transition. There is a critical value ρc ∈ (0,∞] characterized by the rate kernel. For ρ ≤ ρc, there exists a unique equilibrium state ωρ and the solution converges strongly to ωρ. If ρ > ρc the solution converges only weakly to ωρc. In particular, the excess ρ − ρc 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.
|KEYWORDS||convergence to equilibrium, exchange-driven growth, entropy method, mean-field equation, zero-range process|