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Convergence to the equilibrium for the kinetic transport equation in the two-dimensional periodic Lorentz Gas

Francesca Pieroni

TL;DR

The paper analyzes the Boltzmann-Grad limit for the 2D periodic Lorentz gas and analyzes the long-time relaxation of the kinetic density on an extended phase space that includes the time to the next collision $s$ and the collision impact parameter $h$. Using a Fourier-analytic decomposition in space and a fixed-point approach for multi-collision contributions, the authors prove convergence of the density to the equilibrium state $\frac{\langle\mu_0\rangle}{2\pi}E$ in $L^p$, with $^*$-weak convergence for $p=\infty$, and obtain decay rates $\sim 1/(t+1)$ when $p<\infty$ or when the initial data is $x$-independent; for $p=2$ they provide sharper decay bounds. The core method hinges on representing the time evolution via kernels $Q$, their iterates $Q^{(n)}$, and associated invariants $E^{(n)}$, together with analysis of the Fourier modes $\mu_t^k$ and the auxiliary functions $\varphi,\varphi^k$ that encode higher-collision contributions. The results yield convergence on the torus (and weak convergence on $\mathbb{R}^2$) and deepen the kinetic theory understanding of periodic Lorentz gases by quantifying how memory effects and long free paths dissipate over time.

Abstract

We consider the Boltzmann-Grad limit of the two-dimensional periodic Lorentz Gas. It has been proved in [6,14,4] that the time evolution of a probability density on $\mathbb{R}^2\times\mathbb{T}^1\ni(x,v)$ is obtained by extending the phase space $\mathbb{R}^2\times\mathbb{T}^1$ to $\mathbb{R}^2\times\mathbb{T}^1\times[0,+\infty)\times[-1,1]$, where $s\in[0,+\infty)$ represents the time to the next collision and $h\in[-1,1]$ the corresponding impact parameter. Here we prove that under suitable conditions the time evolution of an initial datum in $L^p(\mathbb{T}^2\times\mathbb{T}^1\times[0,+\infty)\times[-1,1])$ converges to the equilibrium state with respect to the $L^p$ norm ($^*$-weakly if $p=\infty$). If $p=2$, or if the initial datum does not depend on $x$, we also get more precise estimates about the rate of the approach to the equilibrium. Our proof is based on the analysis of the long time behavior of the Fourier coefficients of the solution.

Convergence to the equilibrium for the kinetic transport equation in the two-dimensional periodic Lorentz Gas

TL;DR

The paper analyzes the Boltzmann-Grad limit for the 2D periodic Lorentz gas and analyzes the long-time relaxation of the kinetic density on an extended phase space that includes the time to the next collision and the collision impact parameter . Using a Fourier-analytic decomposition in space and a fixed-point approach for multi-collision contributions, the authors prove convergence of the density to the equilibrium state in , with -weak convergence for , and obtain decay rates when or when the initial data is -independent; for they provide sharper decay bounds. The core method hinges on representing the time evolution via kernels , their iterates , and associated invariants , together with analysis of the Fourier modes and the auxiliary functions that encode higher-collision contributions. The results yield convergence on the torus (and weak convergence on ) and deepen the kinetic theory understanding of periodic Lorentz gases by quantifying how memory effects and long free paths dissipate over time.

Abstract

We consider the Boltzmann-Grad limit of the two-dimensional periodic Lorentz Gas. It has been proved in [6,14,4] that the time evolution of a probability density on is obtained by extending the phase space to , where represents the time to the next collision and the corresponding impact parameter. Here we prove that under suitable conditions the time evolution of an initial datum in converges to the equilibrium state with respect to the norm (-weakly if ). If , or if the initial datum does not depend on , we also get more precise estimates about the rate of the approach to the equilibrium. Our proof is based on the analysis of the long time behavior of the Fourier coefficients of the solution.

Paper Structure

This paper contains 40 sections, 21 theorems, 325 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The periodic Lorentz Gas
  3. The distribution of the free path length in the Boltzmann-Grad limit.
  4. The kinetic theory for the periodic Lorentz gas in the Boltzmann-Grad limit.
  5. Averaging on the position $x$.
  6. Averaging on both the position $x$ and the velocity $v(\theta)$.
  7. Further literature.
  8. Main results.
  9. Outline of the paper.
  10. Notations and Definitions.
  11. Notations.
  12. Definitions.
  13. The long time evolution of a density depending on $(\theta,s,h)$.
  14. Writing $\mu_t(\theta,0,h)$ as a linear function of $\mu_0$.
  15. Proof of Proposition \ref{['prop:musvarphi']}.
  16. ...and 25 more sections

Key Result

Theorem 1.1

There exists a constant $C>0$ depending only on $Q$ such that for any $p\in[1,+\infty]$ and $\mu_0\in L^1\cap L^p(\mathbb{T}^1_{2\pi}\times[0,+\infty)\times[-1,1])$, if $\mu_t$ is the solution of the equation eq:ev_x with initial datum $\mu_0$, then In particular, if $\mu_0(\theta,s,h)=\mu_{in}(\theta)E(s,h)$ with $\mu_{in}\in L^p(\mathbb{T}^2\times\mathbb{T}^1_{2\pi})$, then $\mu_0\in L^1\cap L^

Figures (1)

  • Figure B.2.1: The support of $E$ is defined by the curve $\{s\geq1,s=\frac{1}{1-|h|}\}\cup\{s=0,h\in[-1,1]\}\cup\{s\geq0,h=\pm1\}$.

Theorems & Definitions (49)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • ...and 39 more