A sharp commutator estimate for all Riesz modulated energies

2025/11/17 by Elias Hess-Childs, Hess-Childs, Elias, Matthew Rosenzweig +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #26D10 #35J70 #35Q35 #35Q70 #82C22 #82C70 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Gas Dynamics and Kinetic Theory #Mathematical Physics (math-ph) #Probability (math.PR) #Spectral Theory in Mathematical Physics #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2511.13461

openalex publication_date 2025/11/17 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28

Abstract

We prove a functional inequality in any dimension controlling the derivative along a transport of the Riesz modulated energy in terms of the modulated energy itself. This modulated energy was introduced by the third author and collaborators in the study of mean-field limits and statistical mechanics of Coulomb/Riesz gases, where this control is an essential ingredient. Previous work of the last two authors and Q.H. Nguyen arXiv:2107.02592 showed a similar functional inequality but with an additive N-dependent error (where N is the number of particles, d the dimension, and s the inverse power of the Riesz potential) which was not sharp. In this paper, we obtain the optimal N^(s)/(d)-1 error, for all cases, including the sub-Coulomb case. Our method is conceptually simple and, like previous work, relies on the observation that the derivative along a transport of the modulated energy is the quadratic form of a commutator. Through a new potential truncation scheme based on a wavelet-type representation of the Riesz potential to handle its singularity, the proof reduces to averaging over a family of Kato-Ponce type estimates. The commutator estimate has applications to sharp rates of convergence for mean-field limits, quasi-neutral limits, and central limit theorems for the fluctuations of Coulomb/Riesz gases both at and out of thermal equilibrium. In particular, we show here for s<d-2 the expected N^(s)/(d)-1-rate in the modulated energy distance for the mean-field convergence of first-order Hamiltonian and gradient flows. This complements the recent work arXiv:2407.15650 on the optimal rate for the (super-)Coulomb case d-2≤ s<d and therefore resolves the entire potential Riesz case.

Citations

• Correlation estimates for Brownian particles with singular interactions
• Convergence rate for Fluctuations of mean field interacting diffusion and application to 2D viscous Vortex model and Coulomb potential
• Singular flows with time-varying weights
• Propagation of Chaos for 2D Log Gas on the Whole Space
• The Lake equation as a supercritical mean-field limit
• Lectures on Coulomb and Riesz gases
• Sharp commutator estimates of all order for Coulomb and Riesz modulated energies
• Sharp local propagation of chaos for mean field particles with W^-1,∞ kernels
• Relative entropy and modulated free energy without confinement via self-similar transformation
• A duality method for mean-field limits with singular interactions
• Large deviation principles for singular Riesz-type diffusive flows
• The attractive log gas: stability, uniqueness, and propagation of chaos
• Quantitative Propagation of Chaos for 2D Viscous Vortex Model on the Whole Space
• Modulated logarithmic Sobolev inequalities and generation of chaos
• Generative Sliced MMD Flows with Riesz Kernels
• Sharp uniform-in-time mean-field convergence for singular periodic Riesz flows
• Fractional operators as traces of operator-valued curves
• Sharp uniform-in-time propagation of chaos
• Mean-Field Limit Derivation of a Monokinetic Spray Model with Gyroscopic Effects
• Mean-Field Limits in Statistical Dynamics
• From Quantum Many-Body Systems to Ideal Fluids
• Global-in-time mean-field convergence for singular Riesz-type diffusive flows
• Uniform in time propagation of chaos for the 2D vortex model and other singular stochastic systems
• Mean-field limits of Riesz-type singular flows
• On the Rigorous Derivation of the Incompressible Euler Equation from Newton's Second Law
• On the rigorous derivation of the incompressible Euler equation from Newton’s second law
• Classical solutions for fractional porous medium flow
• The Mean-Field Limit of Stochastic Point Vortex Systems with Multiplicative Noise
• Mean-field limit and quantitative estimates with singular attractive kernels
• Dynamics of Particles on a Curve with Pairwise Hyper-singular Repulsion
• From Newton's second law to Euler's equations of perfect fluids
• Gaussian Fluctuations and Free Energy Expansion for Coulomb Gases at Any Temperature
• Modulated Free Energy and Mean Field Limit
• Local laws and rigidity for Coulomb gases at any temperature
• Quantitative Propagation of Chaos in the bimolecular chemical reaction-diffusion model
• Propagation of chaos for a class of first order models with singular mean field interactions
• Fluctuations of Two-Dimensional Coulomb Gases
• Next order energy asymptotics for Riesz potentials on flat tori
• Mean-field limits for some Riesz interaction gradient flows
• Higher Dimensional Coulomb Gases and Renormalized Energy Functionals
• The Kato-Ponce Inequality
• 1D Log Gases and the Renormalized Energy: Crystallization at Vanishing Temperature
• 2D Coulomb Gases and the Renormalized Energy
• A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity
• Extension problem and Harnack's inequality for some fractional operators
• A Kernel Method for the Two-Sample Problem

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• Convergence of Empirical Measures for i.i.d. samples in W^{-α, p}

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• Mean field limit for Coulomb-type flows