Singular Mean-Field Limits via Multiscale Mollification Metrics
Time: 2026-08-28
Published By: Qiuye Huang
Speaker(s): Quoc Hung Nguyen (AMSS, CAS)
Time: 10:00-12:00 September 2, 2026
Venue: Room 78201, Jingchunyuan 78, BICMR
Abstract:
We consider deterministic systems of \(N\) interacting particles in \(\mathbb{R}^d\) driven by singular first-order interactions in the mean-field scaling. The interaction kernel may be attractive, non-symmetric, and need not derive from an energy, allowing forces with inverse-power singularities up to and beyond the Coulomb threshold.
The main difficulty is that classical Wasserstein-type approaches are not strong enough for highly singular kernels, while modulated-energy methods typically rely on repulsive potential structure. We introduce a different strategy based on a multiscale mollification metric: the empirical measure is regularized by heat kernels at all scales above the microscopic scale N^{-1/d}, and the evolution of the resulting weighted \(L^\infty\)-type distance is controlled through a cascade of estimates from finer to coarser scales.
This method yields quantitative convergence of the empirical measure to the solution of the mean-field equation in the sub-Coulomb regime, and also in the one- and two-dimensional Coulomb cases, where a weak notion of particle continuation beyond collisions is introduced. In the higher-dimensional Coulomb case, the convergence holds up to a short time independent of N. In the super-Coulomb regime, the estimates hold on the natural rescaled time scale N^{-1/d}, matching the strength of the singular interaction.
Finally, we show that these time scales are sharp: for a class of attractive interactions, particle collisions occur within the same time scale. Thus the method not only proves convergence but also identifies the precise obstruction to extending the classical particle dynamics beyond the stated regimes.
We consider deterministic systems of \(N\) interacting particles in \(\mathbb{R}^d\) driven by singular first-order interactions in the mean-field scaling. The interaction kernel may be attractive, non-symmetric, and need not derive from an energy, allowing forces with inverse-power singularities up to and beyond the Coulomb threshold.
The main difficulty is that classical Wasserstein-type approaches are not strong enough for highly singular kernels, while modulated-energy methods typically rely on repulsive potential structure. We introduce a different strategy based on a multiscale mollification metric: the empirical measure is regularized by heat kernels at all scales above the microscopic scale N^{-1/d}, and the evolution of the resulting weighted \(L^\infty\)-type distance is controlled through a cascade of estimates from finer to coarser scales.
This method yields quantitative convergence of the empirical measure to the solution of the mean-field equation in the sub-Coulomb regime, and also in the one- and two-dimensional Coulomb cases, where a weak notion of particle continuation beyond collisions is introduced. In the higher-dimensional Coulomb case, the convergence holds up to a short time independent of N. In the super-Coulomb regime, the estimates hold on the natural rescaled time scale N^{-1/d}, matching the strength of the singular interaction.
Finally, we show that these time scales are sharp: for a class of attractive interactions, particle collisions occur within the same time scale. Thus the method not only proves convergence but also identifies the precise obstruction to extending the classical particle dynamics beyond the stated regimes.
