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Long time validity of the linearized Boltzmann uncut-off and the linearized Landau equations from the Newton Law

Corentin Le Bihan

Abstract

We provide a rigorous justification of the linearized Boltzmann- and Landau equations from interacting particle systems with long-range interaction. The result shows that the fluctuations of Hamiltonian $N$- particle systems governed by truncated power law potentials of the form $\mathcal{U}(r)\sim |r/\varepsilon|^{-s}$ (near $r \approx 0$) converge to solutions of kinetic equations in appropriate scaling limits $\varepsilon \rightarrow 0$ and $N\rightarrow \infty$. We prove that for $s\in [0,1)$, the limiting system approaches the uncutoff linearized Boltzmann equation or the linearized Landau equation, depending on the scaling limit. The Coulomb singularity $s=1$ appears as a threshold value. Kinetic scaling limits with $s\in (0,1]$ universally converge to the linearized Landau equation, and we prove the onset of the Coulomb logarithm for $s=1$. To the best of our knowledge, this is the first result on the derivation of kinetic equations from interacting particle systems with long-range power-law interaction.

Long time validity of the linearized Boltzmann uncut-off and the linearized Landau equations from the Newton Law

Abstract

We provide a rigorous justification of the linearized Boltzmann- and Landau equations from interacting particle systems with long-range interaction. The result shows that the fluctuations of Hamiltonian - particle systems governed by truncated power law potentials of the form (near ) converge to solutions of kinetic equations in appropriate scaling limits and . We prove that for , the limiting system approaches the uncutoff linearized Boltzmann equation or the linearized Landau equation, depending on the scaling limit. The Coulomb singularity appears as a threshold value. Kinetic scaling limits with universally converge to the linearized Landau equation, and we prove the onset of the Coulomb logarithm for . To the best of our knowledge, this is the first result on the derivation of kinetic equations from interacting particle systems with long-range power-law interaction.
Paper Structure (34 sections, 28 theorems, 388 equations, 11 figures)

This paper contains 34 sections, 28 theorems, 388 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Modification of the scaling parameters and the sub-Coulomb case ($s\in[0,1)$)
  3. State of the art
  4. Definition of the system and strategy of the proof
  5. The Hamiltonian dynamics
  6. Grand-canonical ensemble and stationary measure
  7. Binary collision, scattering, and definition of the linearized Boltzmann operator
  8. Convergence to the linearized Boltzmann equation with a cut-off
  9. Derivation of the linearized Landau equation and Boltzmann equation without cut-off
  10. The case where the singularity $\tfrac{1}{r^s}$ of the potential is stronger than the Coulomb singularity ($s>1$).
  11. The Coulomb case $s=1$.
  12. Treat now the weak singularity $s\in[0,1)$.
  13. Strategy of the proof of Theorem \ref{['thm:Borne quantitativeè']}
  14. Development along pseudotrajectories and time sampling
  15. Dynamical cluster development
  16. ...and 19 more sections

Key Result

Proposition 2.2

For any continuous and bounded test function $g:{\mathbb{T}}\times\mathbb{R}^d\to \mathbb{R}$, for all $t\in \mathbb{R}$ and for any $\delta>0$,

Figures (11)

  • Figure 1: The first particle will meet the second one. Here $v$ is of order $1$.
  • Figure 2: The scattering between two particles.
  • Figure 3: Exemple of pseudotrajectory for four particles.
  • Figure 4: An example of one pathological pseudotrajectory (on the left) and a non-pathological one (on the right)
  • Figure 5: On the left a tree pseudotrajectory, on the right a graph pseudotrajectory.
  • ...and 6 more figures

Theorems & Definitions (92)

  • Remark 1.1
  • Claim 1
  • Claim 2
  • Remark 1.2
  • Proposition 2.2
  • Remark 2.1
  • Definition 2.1
  • Remark 2.2
  • Theorem 1
  • Remark 2.3
  • ...and 82 more