Cluster Formation in Diffusive Systems

TL;DR

This paper investigates how particles in stochastic systems form clusters under short-range attraction, using kinetic Langevin dynamics, stability analysis, and numerical methods to connect different dynamical regimes.

Contribution

It introduces a theoretical and numerical framework for understanding cluster formation in low-friction Langevin systems, bridging overdamped and Hamiltonian limits.

Findings

Clusters form at low temperatures and short interaction ranges.

Cluster formation time depends on friction and particle number.

Numerical measurements match theoretical predictions.

Abstract

In this paper, we study the formation of clusters for stochastic interacting particle systems (SIPS) that interact through short-range attractive potentials in a periodic domain. We consider kinetic (underdamped) Langevin dynamics and focus on the low-friction regime. Employing a linear stability analysis for the kinetic McKean-Vlasov equation, we show that, at sufficiently low temperatures, and for sufficiently short-ranged interactions, the particles form clusters that correspond to metastable states of the mean-field dynamics. We derive the friction and particle-count dependent cluster-formation time and numerically measure the friction-dependent times to reach a stationary state (given by a state in which all particles are bound in a single cluster). By providing both theory and numerical methods in the inertial stochastic setting, this work acts as a bridge between cluster formation studies in overdamped Langevin dynamics and the Hamiltonian (microcanonical) limit.