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A model for slowing particles in random media

François Golse, Valeria Ricci, Ana Jacinta Soares

TL;DR

This work analyzes the slow-down dynamics of point particles moving through a random medium of small spherical obstacles arranged via a Poisson process. By a Boltzmann-Grad scaling with obstacle radius $\epsilon$ and intensity $\lambda_\epsilon=\lambda/\epsilon^{d-1}$, the authors derive a limit kinetic equation for the particle density that acts on speed only and incorporates a mass-preserving Dirac delta at zero velocity to balance the collision-induced losses. The main technical tool is a semiexplicit backward-flow representation together with a flow-tube decomposition and the Markovian set $A^{\epsilon}_1$, which allow control of obstacle interactions and lead to a well-posed limit equation with an explicit collision kernel depending on the slowing profile $S(|v|)$ and the inverse function $a(z)=\int_0^z \frac{du}{S(u)}$. The results have potential relevance for modeling energy deposition and stopping in random media, including inertial confinement fusion contexts, where slow-down and mass conservation are critical features of the macroscopic transport description.

Abstract

We present a simple model in dimension $d\geq 2$ for slowing particles in random media, where point particles move in straight lines among and inside spherical identical obstacles with Poisson distributed centres. When crossing an obstacle, a particle is slowed down according to the law $\dot{V}= -\fracκε S(|V|) V$, where $V$ is the velocity of the point particle, $κ$ is a positive constant, $ε$ is the radius of the obstacle and $S(|V|)$ is a given slowing profile. With this choice, the slowing rate in the obstacles is such that the variation of speed at each crossing is of order $1$. We study the asymptotic limit of the particle system when $ε$ vanishes and the mean free path of the point particles stays finite. We prove the convergence of the point particles density measure to the solution of a kinetic-like equation with a collision term which includes a contribution proportional to a $δ$ function in $v=0$; this contribution guarantees the conservation of mass for the limit equation.

A model for slowing particles in random media

TL;DR

This work analyzes the slow-down dynamics of point particles moving through a random medium of small spherical obstacles arranged via a Poisson process. By a Boltzmann-Grad scaling with obstacle radius and intensity , the authors derive a limit kinetic equation for the particle density that acts on speed only and incorporates a mass-preserving Dirac delta at zero velocity to balance the collision-induced losses. The main technical tool is a semiexplicit backward-flow representation together with a flow-tube decomposition and the Markovian set , which allow control of obstacle interactions and lead to a well-posed limit equation with an explicit collision kernel depending on the slowing profile and the inverse function . The results have potential relevance for modeling energy deposition and stopping in random media, including inertial confinement fusion contexts, where slow-down and mass conservation are critical features of the macroscopic transport description.

Abstract

We present a simple model in dimension for slowing particles in random media, where point particles move in straight lines among and inside spherical identical obstacles with Poisson distributed centres. When crossing an obstacle, a particle is slowed down according to the law , where is the velocity of the point particle, is a positive constant, is the radius of the obstacle and is a given slowing profile. With this choice, the slowing rate in the obstacles is such that the variation of speed at each crossing is of order . We study the asymptotic limit of the particle system when vanishes and the mean free path of the point particles stays finite. We prove the convergence of the point particles density measure to the solution of a kinetic-like equation with a collision term which includes a contribution proportional to a function in ; this contribution guarantees the conservation of mass for the limit equation.
Paper Structure (12 sections, 9 theorems, 129 equations)

This paper contains 12 sections, 9 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. The particle model and the main result
  3. A priori estimates and considerations about limit points
  4. The limit equation
  5. The backward flow
  6. The Jacobian $J_{{\epsilon},C}(t,x,v)$
  7. The flow tube
  8. The Markovian set $A^{\epsilon}_1(x,v)$ and the limit dynamics
  9. Irrelevance of $(A^{\epsilon}_1(x,v))^c$ when ${\epsilon}\to 0^+$
  10. The dynamics on $A^{\epsilon}_1(x,v)$
  11. The limit dynamics on $[A^{\epsilon}_1(t,x,v)]$
  12. Limit of ${\Bbb E}[f_{\epsilon}(t,x,v){\bf 1}_{|v|>0} {\bf 1}_{A^{\epsilon}_1}]$

Key Result

Theorem 1

Assume $f_0\in C({\Bbb R}^d\times{\Bbb R}^d)$, with compact support and $f_0\geq 0$, and let $f_{\epsilon} (t,x,v;C)$ be the solution to equation (evomisura) with initial datum $f_0$. Let $a(z)=\int_{0}^{z} du \frac{1}{S(u)}$, with $S\in \operatorname{Lip}((0,+\infty))$, satisfying $S\geq S_0>0$. Th with $|z|= a^{-1}(a(|v|)+2 {\kappa}\sqrt{1-h^2})$, $z=|z|\hat{v}$, ${\sigma}=\lambda B^{d-1}$ and

Theorems & Definitions (21)

  • Theorem 1
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Remark 1
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • ...and 11 more