# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1803.04017