Particles approximation for some 1D kinetic Fokker-Planck equations with singular forces

TL;DR

This paper establishes a quantitative convergence of a particle system with singular interactions to a Vlasov-Fokker-Planck equation, using entropy dissipation and energy control techniques.

Contribution

It introduces a novel approach to analyze particle approximations for 1D kinetic equations with singular forces, providing explicit convergence estimates.

Findings

Quantitative convergence estimates for particle systems with singular forces.

Effective control of entropy dissipation and mechanical energy.

Validation of the particle approximation to the limiting PDE.

Abstract

In this paper we consider a system of $N$ particles on the real line evolving according to Newton's law, interacting through a singular (repulsive) force deriving from the potential $\frac{|x|^{1-\alpha}}{1-\alpha}$ with $\alpha \in (0,1/8)$ and Brownian force. Thanks to the entropy dissipation along the Liouville equation associated to this particles system and a control of the mechanic energy, we establish a quantitative estimate which enables to conclude to a convergence/consistency the particles system toward the limiting Vlasov-Fokker-Planck equation with singular force.