# On the mean-field limit for the Vlasov-Poisson-Fokker-Planck system

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Abstract

We devise and study a random particle blob method for approximating the
Vlasov-Poisson-Fokkker-Planck (VPFP) equations by a $N$-particle system subject
to the Brownian motion in $\mathbb{R}^3$ space. More precisely, we show that
maximal distance between the exact microscopic and the mean-field trajectories
is bounded by $N^{-\frac{1}{3}+\varepsilon}$
($\frac{1}{63}\leq\varepsilon<\frac{1}{36}$) for a system with blob size
$N^{-\delta}$ ($\frac{1}{3}\leq\delta<\frac{19}{54}-\frac{2\varepsilon}{3}$) up
to a probability $1-N^{-\alpha}$ for any $\alpha>0$, which improves the cut-off
in [10]. Our result thus leads to a derivation of VPFP equations from the
microscopic $N$-particle system. In particular we prove the convergence rate
between the empirical measure associated to the particle system and the
solution of the VPFP equations. The technical novelty of this paper is that our
estimates crucially rely on the randomness coming from the initial data and
from the Brownian motion.

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https://hdl.handle.net/10161/17119Collections

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Show full item record### Scholars@Duke

#### Peter Pickl

Visiting Professor of Global Studies

Starting with the autumn term 2018 I will teach the foundational mathematics and integrated
science courses in the undergraduate program at DKU. In the coming years, other classes
on several topics of mathematics and mathematical physics will be taught.

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