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The Lorentz Gas in a Mean-Field Potential: Weak Coupling and Diffusive Regime

Dominik Nowak

Abstract

We investigate the diffusive scaling of the Lorentz gas in the presence of an external force of mean-field type. In the weak coupling regime and for diffusive time scales, the test particle's law converges to the probability density satisfying the heat equation. The diffusion coefficient of the heat equation is given by the Green-Kubo relation.

The Lorentz Gas in a Mean-Field Potential: Weak Coupling and Diffusive Regime

Abstract

We investigate the diffusive scaling of the Lorentz gas in the presence of an external force of mean-field type. In the weak coupling regime and for diffusive time scales, the test particle's law converges to the probability density satisfying the heat equation. The diffusion coefficient of the heat equation is given by the Green-Kubo relation.

Paper Structure

This paper contains 4 sections, 4 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Model and Notation
  3. Result
  4. Proof of Theorem \ref{['thm:HeatEquation']}

Key Result

Theorem 1

Let $f_0 \in C_0(\mathbb{R}^d \times \mathbb{R}^d)$ be a compactly supported initial probability density. Suppose that $f_0$ has two bounded derivatives with respect to $x$ and $v$, then we prove that for all $t \in (0,T]$ with $T >0$ in $L^2(\mathbb{R}^d\times S^{d-1}_\abs{v} )$ and $K^{-1} := \frac{2 \pi^{d/2}}{\Gamma(d/2) \abs{v}^{d-1}}$. Furthermore, if we investigate higher densities $\mu_\v

Theorems & Definitions (10)

  • Theorem 1
  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 2
  • Proposition 3
  • proof
  • Remark 3