Cluster Formation in Diffusive Systems

TL;DR

This work analyzes cluster formation in weakly interacting, inertial (underdamped) Langevin systems on a periodic domain, focusing on low-temperature regimes where attractive interactions induce metastable clusters. It combines a mean-field kinetic McKean-Vlasov PDE with a detailed linear stability analysis, revealing a friction-independent critical temperature $\beta_c$ that governs the onset of clustering and a friction-dependent timescale for cluster formation. A spectral approach is developed to efficiently compute the linear growth rates, and extensive numerical simulations using dedicated IPS and DBSCAN-based cluster detection validate the theory in 1D and 2D and explore the Hamiltonian limit. The results bridge overdamped and Hamiltonian limits, quantify metastability, and provide practical tools for predicting clustering times and critical temperatures in inertial stochastic particle systems.

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.

Figures (29)

• Figure 1: Example trajectories for the one-dimensional interacting particle system starting from a uniform distribution. Left: for reciprocal temperature $\beta<\beta_c$ the uniform distribution remains stable. Right:$\beta>\beta_c$ (i.e., small enough temperatures) the particle trajectories form two clusters that later merge into one cluster, yielding the new stationary state.
• Figure 2: Visualization of the IPS evolution in time in one dimension (left) and two dimensions (right) at $\beta > \beta_c$, leading to cluster formation. The clusters merge until only one cluster is left.
• Figure 3: Convergence of $\psi_{\max}$ obtained from $\boldsymbol{A}(k,\beta)$ when considering increasing numbers of matrix dimensions (left) and wavenumbers (right). Ground truth estimate $\hat{\psi}_{\max}$ was obtained by truncating after 100 matrix rows and using 30 wavenumbers (plus their negative images). Interaction potential \ref{['eq: wrapped_gaussian']} for $\sigma = 1/\sqrt{2}$, $L = 10$, $\beta = 25 > \beta_{c}$ and $\gamma = 1.0$.
• Figure 4: The relationship between friction, inverse temperature and $\psi_{\max}$, illustrating a discontinuous phase transition at $\beta_{c}$. Left: Gaussian potential for $\sigma = 1/\sqrt{2}$ and $L = 10$. Middle: Morse potential for $a = 1$, $D_{e} = 2$ and $L = 10$. Right: GEM-$\alpha$ potential for $\sigma = 1/\sqrt{2}$, $\alpha =4$ and $L = 10$.
• Figure 5: $\mathbb{E}\left[\rho_{1}(t,x)\overline{\rho_{1}}(t,x')\right]$ from \ref{['eq:fourier_sum']} for the Gaussian potential for $\sigma = 1/\sqrt{2}$, $L = 10$, $\gamma = 1.0$ and $\beta = 25 > \beta_{c}$.