# American Institute of Mathematical Sciences

June  2018, 11(3): 647-695. doi: 10.3934/krm.2018027

## The rigorous derivation of the Linear Landau equation from a particle system in a weak-coupling limit

 Dipartimento di Matematica, Università di Roma "La Sapienza", P.le A. Moro, 5, 00185 Roma, Italy

* Corresponding author: Nicolo' Catapano

Received  January 2017 Revised  September 2017 Published  March 2018

We consider a system of N particles interacting via a short-range smooth potential, in a weak-coupling regime. This means that the number of particles $N$ goes to infinity and the range of the potential $ε$ goes to zero in such a way that $Nε^{2} = α$, with $α$ diverging in a suitable way. We provide a rigorous derivation of the Linear Landau equation from this particle system. The strategy of the proof consists in showing the asymptotic equivalence between the one-particle marginal and the solution of the linear Boltzmann equation with vanishing mean free path. This point follows [3] and makes use of technicalities developed in [16]. Then, following the ideas of Landau, we prove the asympotic equivalence between the solutions of the Boltzmann and Landau linear equation in the grazing collision limit.

Citation: Nicolo' Catapano. The rigorous derivation of the Linear Landau equation from a particle system in a weak-coupling limit. Kinetic & Related Models, 2018, 11 (3) : 647-695. doi: 10.3934/krm.2018027
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##### References:
We denote with $\sigma\in S^{2}\left(\frac{v_{1}+v_{2}}{2}\right)$ the direction of $V^{'}$ and with $\theta$ the angle between $V$ and $V^{'}$
Here $\omega = \omega(\nu, V)$ is the unit vector bisecting the angle between $-V$ and $V'$, $\nu$ is the unit vector pointing from the particle with velocity $v_{1}$ to the particle with velocity $v_{2}$ when they are about to collide. We denote with $\beta$ the angle between $-V$ and $\omega$, with $\varphi$ the angle between $-V$ and $\nu$, with $\rho = \sin\varphi$ the impact parameter and with $\theta$ the deflection angle. It results that $\theta = \pi-2\beta$
A representation of a three dimensional scattering
A representation of the tree graph $(1, 1, 2)$. At the time $t_{1}$ we create the particle $2$ on the particle $1$. Then at time $t_{2}$ we create the particle $3$ on the particle $1$. Finally at time $t_{3}$ the particle $4$ is created on the particle $2$
We used a dashed line to evidence the virtual trajectory of the fourth particle
The virtual trajectory of the praticles $i$ and $k$ and their backward history
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