Formation of clusters and coarsening in weakly interacting diffusions

TL;DR

This work analyzes clustering in weakly interacting diffusions with localized attractive potentials on the 1D torus and links cluster formation to discontinuous phase transitions in the McKean–Vlasov PDE. It develops a variational framework showing global minimizers are either uniform or single-cluster via a variant of the strict Riesz rearrangement inequality, and introduces a reduced mass-exchange model that captures dynamical metastability of the mean-field limit. A coarse-grained description of coalescing heavy Brownian motions with mass exchange provides insight into coarsening mechanisms and timescales, supported by numerical experiments. The results illuminate how mean-field dynamics fail to capture cluster coalescence, while a mass-exchange perspective explains long-lived multi-cluster states and their abrupt collapse, with potential links to Massive Arratia-type limits in suitable scalings.

Abstract

This paper studies the clustering behavior of weakly interacting diffusions under the influence of sufficiently localized attractive interaction potentials on the one-dimensional torus. We describe how this clustering behavior is closely related to the presence of discontinuous phase transitions in the mean-field PDE. For local attractive interactions, we employ a new variant of the strict Riesz rearrangement inequality to prove that all global minimizers of the free energy are either uniform or single-cluster states, in the sense that they are symmetrically decreasing. We analyze different timescales for the particle system and the mean-field (McKean-Vlasov) PDE, arguing that while the particle system can exhibit coarsening by both coalescence and diffusive mass exchange between clusters, the clusters in the mean-field PDE are unable to move and coarsening occurs via the mass exchange of clusters. By introducing a new model for this mass exchange, we argue that the PDE exhibits dynamical metastability. We conclude by presenting careful numerical experiments that demonstrate the validity of our model.

Figures (10)

• Figure 1.1: Initial clustering for the particle model (left) and for the mean-field PDE (right) for the potential \ref{['eq:w-def:Hegselmann--Krause-special']} with $\ell = 0.1$ and $\gamma = 10^4$. For the particle model, initial positions are evenly spaced on $[0,1)$, that is $X^i_0 = \frac{i}{N}$ for $i=1,\dots,N$. The PDE is started from the state $\frac{100}{98}\chi_{[0.01, 0.99]}$ which is a small perturbation of the uniform state. The solution to the PDE is simulated using the Scharfetter--Gummel scheme SS22.
• Figure 1.2: Coalescence of clusters in the particle system \ref{['eq:microscopic-system']} where $N=50$, $\gamma=2000$, $\ell=0.08$ and the particles start at $X^i_0=\frac{i}{N}$. \ref{['eq:microscopic-system']} was simulated using the Euler--Maruyama scheme with time step $\Delta t = 0.00005$. The bottom plot shows the interaction energy $\frac{1}{2N^2}\sum_{i,j=1}^N W_{\gamma,\ell}(X^i_t - X^j_t)$ as an order parameter.
• Figure 1.3: Exchange of mass between three clusters in the mean-field PDE. The potential is given by \ref{['eq:w-def:Hegselmann--Krause-special']} with $\gamma=1000$ and $\ell=0.05$. The initial state is given by a Gaussian mixture (truncated at $0$ and $1$ and normalized) with centers $(0.1, 0.3, 0.7)$ and variances $\sigma_k^2 = \frac{\ell}{\gamma m_k}$ for $k=1,2,3$ and $m_1=0.2, m_2=0.3, m_3=0.5$. This form of the variances will be explained in more detail \ref{['sec:stationary-states:approximate-gaussian']}.
• Figure 1.4: Dynamical metastability in the PDE: The PDE is started from a Gaussian mixture with centers $0.25$ and $0.75$ and initial masses $0.45$, $0.55$, respectively. $w$ is given by \ref{['eq:w-def:Hegselmann--Krause-special']} and $\gamma=1000$ and $\ell=0.05$.
• Figure 3.1: Bifurcation model for the Hegselmann--Krause potential \ref{['eq:w-def:Hegselmann--Krause-special']}. Holding $\ell=0.1$ fixed, the bifurcation diagram shows the single-cluster state $\rho_\gamma$ as a function of $\gamma$, continuing the single-cluster branch until it ceases to exist. The stationary state $\rho_\gamma$ is computed by using Newton's method to find fixed points of the Kirkwood--Monroe map. The blue curve is the free energy gap $\mathcal{F}_{\gamma,\ell}(\rho_\gamma) - \mathcal{F}_{\gamma,\ell}(\rho_\infty)$, where $\rho_\infty\equiv 1$ is the uniform stationary state. The red curve shows the $L^1$-norm of $\rho_\gamma - \rho_\infty$. At $\gamma = \gamma_c$, there is a discontinuous phase transition. The uniform state $\gamma_\infty$ for $\gamma\in[0,\gamma_\sharp)$ where $\gamma_\sharp$ is the point of linear stability. Note that the single-cluster branch exists for $\gamma<\gamma_c$ close to $\gamma_c$, however, the branch is the global minimizer only for $\gamma > \gamma_c$.

Theorems & Definitions (16)

• Definition 3.1: Transition point
• Corollary 3.2
• proof
• Theorem 3.3: Global minimizers of the free energy are symmetrically decreasing
• Corollary 3.4: Single-cluster minimizer
• Theorem 3.5: Riesz rearrangement inequality
• Lemma 3.6: Variant of the strict Riesz rearrangement inequality
• proof
• Lemma 3.7: Analyticity of stationary states
• proof