Metastable States for an Aggregation Model with Noise

We study the long-time effect of noise on pattern formation for the aggregation model. We consider aggregation kernels that generate patterns consisting of two delta-concentrations. Without noise, there is a one-parameter family of admissable equilibria that consists of two concentrations having whose mass is not necessary equal. We show that when small amount of noise is added, the heavier concentration “leaks” its mass towards the lighter concentration over a very long time scale, eventually resulting in the equilibration of the two masses. We use exponentially small asymptotics to derive the long-time ODE’s that quantify this mass exchange. Our theory is validated using full numerical simulations of the original model – both of the original stochastic particle system and its PDE limit. Our formal computations show that adding noise destroys the degeneracy in equilibrium solution and leads to a unique symmetric steady state.