Sharp uniform-in-time mean-field convergence for singular periodic Riesz flows

TL;DR

This paper establishes sharp uniform-in-time mean-field convergence for singular Riesz flows on the torus, combining relaxation rates, modulated free energy, and entropy methods to address a previously unhandled range of parameters.

Contribution

It proves uniform-in-time mean-field convergence with sharp rates for singular Riesz flows on the torus, extending previous results to a broader parameter range and combining multiple analytical techniques.

Findings

Proved global well-posedness and relaxation to equilibrium for the limiting PDE.

Established sharp uniform-in-time mean-field convergence for gradient flows.

Provided an alternative proof of propagation of chaos using entropy methods.

Abstract

We consider conservative and gradient flows for $N$-particle Riesz energies with mean-field scaling on the torus $\mathbb{T}^d$, for $d\geq 1$, and with thermal noise of McKean-Vlasov type. We prove global well-posedness and relaxation to equilibrium rates for the limiting PDE. Combining these relaxation rates with the modulated free energy of Bresch et al. and recent sharp functional inequalities of the last two named authors for variations of Riesz modulated energies along a transport, we prove uniform-in-time mean-field convergence in the gradient case with a rate which is sharp for the modulated energy pseudo-distance. For gradient dynamics, this completes in the periodic case the range $d-2\leq s<d$ not addressed by previous work of the second two authors. We also combine our relaxation estimates with the relative entropy approach of Jabin and Wang for so-called $\dot{W}^{-1,\infty}$ kernels, giving a proof of uniform-in-time propagation of chaos alternative to Guillin et al.