Noisy nonlocal aggregation model with gradient flow structures

TL;DR

This work develops a coherent microscopic–macroscopic framework for noisy nonlocal aggregation by deriving a continuum density equation with a gradient-flow structure in the $W_2$ space and an energy functional $\mathcal{E}[\rho]$ that combines nonlocal interactions, external forces, and entropy. Through variational analysis, it establishes first- and second-order conditions for energy minimizers and demonstrates that these minimizers correspond to the long-time states of the PDE, with noise regularizing the density and preventing singular concentrations. The authors implement Euler–Maruyama and pseudo-spectral numerical schemes to compare particle and density dynamics in 1D and 2D, and compute energy minimizers via Newton iterations, including special cases where $\Theta$ is a Green’s function, reducing to differential equations. Across three representative potentials (noisy HK, Morse-type, and Laplace), they show that energy minimizers align with PDE equilibria and confirm spectral stability, highlighting diffusion’s role in stabilizing and localizing density structures. The framework provides a versatile tool for understanding how stochastic fluctuations interact with nonlocal interactions to shape collective dynamics in biological and social systems.

Abstract

Interacting particle systems provide a fundamental framework for modeling collective behavior in biological, social, and physical systems. In many applications, stochastic perturbations are essential for capturing environmental variability and individual uncertainty, yet their impact on long-term dynamics and equilibrium structure remains incompletely understood, particularly in the presence of nonlocal interactions. We investigate a stochastic interacting particle system governed by potential-driven interactions and its continuum density formulation in the large-population limit. We introduce an energy functional and show that the macroscopic density evolution has a gradient-flow structure in the Wasserstein-2 space. The associated variational framework yields equilibrium states through constrained energy minimization and illustrates how noise regulates the density and mitigates singular concentration. We demonstrate the connection between microscopic and macroscopic descriptions through numerical examples in one and two dimensions. Within the variational framework, we compute energy minimizers and perform a linear stability analysis. The numerical results show that the stable minimizers agree with the long-time dynamics of the macroscopic density model.

Figures (4)

• Figure 1: (a,b) The space-time density evolution for the 1D HK model for the discrete density $\rho^d$ in \ref{['eq: rhod']} and the density $\rho$ from the PDE model \ref{['eq: pde_model']}, respectively. (c) The initial density $\rho_0$, asymptotic densities $\rho_\infty,\rho_\infty^d$ (practically reached at $t=50$), and the energy minimizer $\rho^*$.
• Figure 2: (a,b) Iso-surface plots for the discrete density $\rho^d$ from the SDE model and the density $\rho$ from the PDE 2D HK model at time $t=50$, where the asymptotic limit has been practically reached. In the simulations, we use parameters $c = 0.25$ and $\sigma = 0.05$. (c) The cross section of the densities along the $x_2 = 0$ direction at $t=0$ and $t=50$ and the energy minimizer $\rho^*$ computed by solving \ref{['eq: minimizer equation']}.
• Figure 3: Density evolution for the model with Morse potentials \ref{['eq: Morse kernel']} for (a,d) the spreading case, (b,e) the steady-state case, and (c,f) the blow-up case, where the top panels show the space-time evolution while the bottom panels show the densities at specific times.
• Figure 4: Long-term densities of \ref{['eq: pde_model']} and energy minmizers solved from \ref{['eq: condition1a']} and \ref{["eq: differential eqn by Green's function"]} for parameters (a) $\sigma=0.2$, $\gamma = 0.2$, (b) $\sigma = 0.2$, $\gamma = 0.3$, and (c) $\sigma =0.3$, $\gamma = 0.3$.