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Boltzmann-Grad limit for the inelastic Lorentz gas: Part I. Existence, uniqueness, and rigorous derivation via weak convergence

Théophile Dolmaire, Alessia Nota

TL;DR

This work establishes a rigorous weak-kinetic derivation of the inelastic linear Boltzmann equation from a dissipative Lorentz gas in the Boltzmann-Grad limit with Poisson-distributed scatterers. A central challenge is the loss of energy in collisions, which complicates backward-in-time dynamics and recollisions; the authors overcome this by formulating the problem in a weak framework and exploiting a novel series representation for both the forward solution and the adjoint equation. They prove the existence and uniqueness of weak solutions to the inelastic Boltzmann equation with initial data possessing finite exponential moments, and derive an explicit convergence rate $\varepsilon^{1/4}$ for the microscopic solution $f_\varepsilon$ to the macroscopic limit $f$. They also construct strong solutions via a convergent series under the same moment assumptions. The results provide a quantitative, rigorous bridge from deterministic microscopic dynamics to a dissipative kinetic description, with explicit control on the impact of recollisions and interference events through geometric and probabilistic estimates.

Abstract

In this paper we provide a rigorous derivation of the inelastic linear Boltzmann equation, in the Boltzmann-Grad limit, from a dissipative, random, Lorentz gas in arbitrary dimensions d $\geq$ 2. Specifically, we consider a microscopic particle system where scatterers are randomly distributed according to a Poisson process, and a tagged light particle undergoes inelastic collisions with the scatterers following a reflection law characterized by a fixed restitution coefficient. We establish the existence and uniqueness of weak solutions to the inelastic linear Boltzmann equation within the class of non-negative Radon measures, assuming that the initial data has a finite exponential moment. We first show that the forward dynamics of the dissipative particle system is globally defined almost surely and then prove the weak$-*$ convergence of the microscopic solution towards the weak solutions of the inelastic linear Boltzmann equation, providing an explicit rate of convergence. Furthermore, under the same initial data assumptions, we prove the existence of strong solutions to the inelastic linear Boltzmann equation, constructed via a series representation of the solutions.

Boltzmann-Grad limit for the inelastic Lorentz gas: Part I. Existence, uniqueness, and rigorous derivation via weak convergence

TL;DR

This work establishes a rigorous weak-kinetic derivation of the inelastic linear Boltzmann equation from a dissipative Lorentz gas in the Boltzmann-Grad limit with Poisson-distributed scatterers. A central challenge is the loss of energy in collisions, which complicates backward-in-time dynamics and recollisions; the authors overcome this by formulating the problem in a weak framework and exploiting a novel series representation for both the forward solution and the adjoint equation. They prove the existence and uniqueness of weak solutions to the inelastic Boltzmann equation with initial data possessing finite exponential moments, and derive an explicit convergence rate for the microscopic solution to the macroscopic limit . They also construct strong solutions via a convergent series under the same moment assumptions. The results provide a quantitative, rigorous bridge from deterministic microscopic dynamics to a dissipative kinetic description, with explicit control on the impact of recollisions and interference events through geometric and probabilistic estimates.

Abstract

In this paper we provide a rigorous derivation of the inelastic linear Boltzmann equation, in the Boltzmann-Grad limit, from a dissipative, random, Lorentz gas in arbitrary dimensions d 2. Specifically, we consider a microscopic particle system where scatterers are randomly distributed according to a Poisson process, and a tagged light particle undergoes inelastic collisions with the scatterers following a reflection law characterized by a fixed restitution coefficient. We establish the existence and uniqueness of weak solutions to the inelastic linear Boltzmann equation within the class of non-negative Radon measures, assuming that the initial data has a finite exponential moment. We first show that the forward dynamics of the dissipative particle system is globally defined almost surely and then prove the weak convergence of the microscopic solution towards the weak solutions of the inelastic linear Boltzmann equation, providing an explicit rate of convergence. Furthermore, under the same initial data assumptions, we prove the existence of strong solutions to the inelastic linear Boltzmann equation, constructed via a series representation of the solutions.

Paper Structure

This paper contains 43 sections, 13 theorems, 326 equations, 2 figures.

Table of Contents

  1. Introduction
  2. State of the art on the Lorentz Gas Model.
  3. The linear inelastic Lorentz model and the inelastic linear Boltzmann equation.
  4. Formal derivation of the inelastic linear Boltzmann equation.
  5. The difficulties concerning the derivation of the inelastic linear Boltzmann equation.
  6. Plan of the paper.
  7. Notations
  8. Model, main results and strategy
  9. The model
  10. The microscopic dynamics
  11. Distributions of scatterers.
  12. The dynamics of the tagged particle.
  13. The Boltzmann-Grad limit
  14. Discussion on the inelastic collapse in the Boltzmann-Grad limit.
  15. Main result
  16. ...and 28 more sections

Key Result

Theorem 1

Let $f_0 \in \mathcal{P}(\mathbb{R}^d\times \mathbb{R}^d) \cap \mathcal{M}_{+,1}(\mathbb{R}^d\times\mathbb{R}^d)$. Let us assume in addition that there exists a constant $p > 1$ such that: and let $f \in \mathcal{C} ([0,+\infty[,\mathcal{M}_+(\mathbb{R}^d\times\mathbb{R}^d))$ be the unique weak solution of the linear inelastic Boltzmann equation EQUATLineaBoltzSphDuFormeForte with initial datum $

Figures (2)

  • Figure 1: Representation of a collision obtained by the scattering mapping: $v' = \kappa_\omega(v)$. In comparison, an elastic collision ($r = 1$) is represented in light grey.
  • Figure 2: Representation of the result of Lemma \ref{['LEMMEEstimMesurTube_Dynam']}. The surface of the twisted tube (in blue) is smaller than the surface of the straight tube (in red), even if the distances $\vert x_1 - x_2 \vert$ and $\vert x_2 - x_3 \vert$ are the same for the left and the right tubes. The phenomenon is independent from the dimension.

Theorems & Definitions (38)

  • Definition 1: Forward flow of the tagged particle
  • Definition 2: Generalized forward flow
  • Remark 1
  • Definition 3: Boltzmann-Grad limit
  • Definition 4: Microscopic solution of the inelastic Lorentz model
  • Definition 5: Weak solution of the linear inelastic Boltzmann equation \ref{['EQUATLineaBoltzSphDuFormeForte']}
  • Theorem 1: Derivation of the weak form of the linear inelastic Boltzmann equation
  • Remark 2
  • Remark 3
  • Theorem 2: Convergence of the series representation in the space of exponential weights in velocity
  • ...and 28 more