Mean-field limits for some Riesz interaction gradient flows

This paper is concerned with the mean-field limit for the gradient flow evolution of particle systems with pairwise Riesz interactions, as the number of particles tends to infinity. Based on a modulated energy method, using regularity and stability properties of the limiting equation, as inspired by the work of Serfaty in the context of the Ginzburg-Landau vortices, we prove a mean-field limit result in dimensions 1 and 2 in cases for which this problem was still open

Mean Field Limit for Coulomb-Type Flows

We establish the mean-field convergence for systems of points evolving along the gradient flow of their interaction energy when the interaction is the Coulomb potential or a super-coulombic Riesz potential, for the first time in arbitrary dimension. The proof is based on a modulated energy method using a Coulomb or Riesz distance, assumes that the solutions of the limiting equation are regular enough and exploits a weak-strong stability property for them. The method can handle the addition of a regular interaction kernel, and applies also to conservative and mixed flows. In the appendix, it is also adapted to prove the mean-field convergence of the solutions to Newton's law with Coulomb or Riesz interaction in the monokinetic case to solutions of an Euler-Poisson type system.Comment: Final version with expanded introduction, to appear in Duke Math Journal. 35 page

Systems of Points with Coulomb Interactions

Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and give rise to a variety of questions pertaining to calculus of variations, Partial Differential Equations and probability. We will review these as well as "the mean-field limit" results that allow to derive effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order beyond the mean-field limit, giving information on the system at the microscopic level. In the setting of statistical mechanics, this allows for instance to observe the effect of the temperature and to connect with crystallization questions.Comment: 30 pages, to appear as Proceedings of the ICM201

Mean-field limits of Riesz-type singular flows with possible multiplicative transport noise

We provide a proof of mean-field convergence of first-order dissipative or conservative dynamics of particles with Riesz-type singular interaction (the model interaction is an inverse power $s$ of the distance for any $0<s<d$) when assuming a certain regularity of the solutions to the limiting evolution equations. It relies on a modulated-energy approach, as introduced in previous works where it was restricted to the Coulomb and super-Coulombic cases. The method also allows us to incorporate multiplicative noise of transport type into the dynamics for the first time in this generality. It relies in extending functional inequalities of arXiv:1803.08345, arXiv:2011.12180, arXiv:2003.11704 to more general interactions, via a new, robust proof that exploits a certain commutator structure.Comment: 40 page

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

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.Comment: 63 page

Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernels

The aim of this paper is twofold. Based on the geometric Wasserstein tangent space, we first introduce Wasserstein steepest descent flows. These are locally absolutely continuous curves in the Wasserstein space whose tangent vectors point into a steepest descent direction of a given functional. This allows the use of Euler forward schemes instead of Jordan--Kinderlehrer--Otto schemes. For $\lambda$-convex functionals, we show that Wasserstein steepest descent flows are an equivalent characterization of Wasserstein gradient flows. The second aim is to study Wasserstein flows of the maximum mean discrepancy with respect to certain Riesz kernels. The crucial part is hereby the treatment of the interaction energy. Although it is not $\lambda$-convex along generalized geodesics, we give analytic expressions for Wasserstein steepest descent flows of the interaction energy starting at Dirac measures. In contrast to smooth kernels, the particle may explode, i.e., a Dirac measure becomes a non-Dirac one. The computation of steepest descent flows amounts to finding equilibrium measures with external fields, which nicely links Wasserstein flows of interaction energies with potential theory. Finally, we provide numerical simulations of Wasserstein steepest descent flows of discrepancies

Non-Linear Langevin and Fractional Fokker-Planck Equations for Anomalous Diffusion by Levy Stable Processes

The~numerical solutions to a non-linear Fractional Fokker--Planck (FFP) equation are studied estimating the generalized diffusion coefficients. The~aim is to model anomalous diffusion using an FFP description with fractional velocity derivatives and Langevin dynamics where L\'{e}vy fluctuations are introduced to model the effect of non-local transport due to fractional diffusion in velocity space. Distribution functions are found using numerical means for varying degrees of fractionality of the stable L\'{e}vy distribution as solutions to the FFP equation. The~statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy and modified transport coefficient. The~transport coefficient significantly increases with decreasing fractality which is corroborated by analysis of experimental data.Comment: 20 pages 7 figure