Metastable states for an aggregation model with noise

TL;DR

This paper investigates how small noise influences pattern formation in an aggregation model, showing that noise causes mass exchange between concentrations, leading to a symmetric equilibrium over long times.

Contribution

The study provides a detailed asymptotic analysis of noise effects on aggregation equilibria, revealing the transition from degenerate to unique symmetric steady states.

Findings

Noise induces mass transfer between concentrations over long times

Adding noise removes degeneracy, resulting in a unique equilibrium

Numerical simulations validate the asymptotic and theoretical predictions

Abstract

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 admissible equilibria that consist of two concentrations whose mass is not necessary equal. We show that when a 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 the equilibrium solution and leads to a unique symmetric steady state.