Uniform-in-Time and Uniform-in-Temperature Particle Approximations of Coulomb MMD Flows

We study particle approximations of maximum mean discrepancy (MMD) flows with the Coulomb kernel on the torus ${\mathbb{T}}^d$, $d\ge 2$, driven by thermal noise of strength $\tau \ge 0$. For smooth, strictly positive initial and target densities and i.i.d. initial positions, we prove that the expected Wasserstein-1 distance between the empirical measure and the mean-field solution is at most of order $N^{-1\mathrm{/}d}$ for $d\ge 3$ and $\sqrt{(\log N)\mathrm{/}N}$ for $d=2$, uniformly in time and in the temperature $\tau \ge 0$, including the deterministic case $\tau =0$. At each fixed $\tau >0$, we also prove propagation of chaos in total variation for any $k$-fold marginal distributions of particles, uniformly in time, with a bound of order $\sqrt{k\mathrm{/}\tau }$ times the empirical convergence rate. Our proof uses the modulated free-energy method of Bresch, Jabin, and Wang to reduce the particle convergence estimate to a relaxation estimate that controls the time integral of the mean-field velocity gradient uniformly in $\tau \ge 0$. Near zero temperature, where the smoothing effect of the noise degenerates, we obtain this estimate by recentering the dynamics at thermal equilibrium and using the damping generated by the Coulomb interaction.