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Fractional diffusion as the limit of a short range potential Rayleigh gas

Karsten Matthies, Theodora Syntaka

TL;DR

The paper rigorously derives a fractional diffusion equation as the hydrodynamic limit of a deterministic Rayleigh gas with a short-range potential in three dimensions. It uses a Boltzmann-Grad scaling with a Poisson background having a fat-tailed velocity distribution and leverages collision-tree semigroup methods to control empirical dynamics and compare them to a linear Boltzmann equation. A Carleman representation and Mellet-type hydrodynamic limit are employed to translate the high-velocity tail into a nonlocal diffusion operator, yielding a fractional diffusion equation of order $\gamma$ on time scales where $cT$ grows like a negative power of the small parameter. The analysis provides quantitative error bounds over extended times and verifies the necessary assumptions to pass from Boltzmann to fractional diffusion, linking micro-scale deterministic dynamics to nonlocal macro-scale transport. This work advances the understanding of nonlocal diffusion arising from deterministic particle systems with heavy-tailed backgrounds.

Abstract

The fractional diffusion equation is rigorously derived as a scaling limit from a deterministic Rayleigh gas, where particles interact via short range potentials with support of size $\varepsilon$ and the background is distributed in space $\mathbb{R}^3$ according to a Poisson process with intensity $N$ and in velocity according to some fat-tailed distribution. As an intermediate step a linear Boltzmann equation is obtained in the Boltzmann-Grad limit as $\varepsilon$ tends to zero and $N$ tends to infinity with $N \varepsilon^2 =c$. The convergence of the empiric particle dynamics to the Boltzmann-type dynamics is shown using semigroup methods to describe probability measures on collision trees associated to physical trajectories in the case of a Rayleigh gas. The fractional diffusion equation is a hydrodynamic limit for times $t \in [0,T]$, where $T$ and inverse mean free path $c$ can both be chosen as some negative rational power $\varepsilon^{-k}$.

Fractional diffusion as the limit of a short range potential Rayleigh gas

TL;DR

The paper rigorously derives a fractional diffusion equation as the hydrodynamic limit of a deterministic Rayleigh gas with a short-range potential in three dimensions. It uses a Boltzmann-Grad scaling with a Poisson background having a fat-tailed velocity distribution and leverages collision-tree semigroup methods to control empirical dynamics and compare them to a linear Boltzmann equation. A Carleman representation and Mellet-type hydrodynamic limit are employed to translate the high-velocity tail into a nonlocal diffusion operator, yielding a fractional diffusion equation of order on time scales where grows like a negative power of the small parameter. The analysis provides quantitative error bounds over extended times and verifies the necessary assumptions to pass from Boltzmann to fractional diffusion, linking micro-scale deterministic dynamics to nonlocal macro-scale transport. This work advances the understanding of nonlocal diffusion arising from deterministic particle systems with heavy-tailed backgrounds.

Abstract

The fractional diffusion equation is rigorously derived as a scaling limit from a deterministic Rayleigh gas, where particles interact via short range potentials with support of size and the background is distributed in space according to a Poisson process with intensity and in velocity according to some fat-tailed distribution. As an intermediate step a linear Boltzmann equation is obtained in the Boltzmann-Grad limit as tends to zero and tends to infinity with . The convergence of the empiric particle dynamics to the Boltzmann-type dynamics is shown using semigroup methods to describe probability measures on collision trees associated to physical trajectories in the case of a Rayleigh gas. The fractional diffusion equation is a hydrodynamic limit for times , where and inverse mean free path can both be chosen as some negative rational power .
Paper Structure (20 sections, 30 theorems, 259 equations, 3 figures)

This paper contains 20 sections, 30 theorems, 259 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Collision Trees
  3. The collision kernel B
  4. Deriving the collision kernel B
  5. Formula of the collision kernel B
  6. Quantitative errors estimates for the Rayleigh gas
  7. Energy estimates
  8. Number of collisions
  9. The set of re-collisions
  10. Initial overlap
  11. The set of binary collisions
  12. Collisions with small relative velocities
  13. Quantitative estimates for bad trees
  14. The Empirical Distribution
  15. Making time $T$ large
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $f_0 \in L^1( \mathds{R}^3 \times \mathds{R}^3)$ the initial distribution of the tagged particle with $f_0(x,v) (1+|v|) \in L^1(\mathds R^3\times \mathds R^3)$ and $g_0 (v) (1+|v|) \in L^1(\mathds{R}^3)$, where $g_0$ is the distribution of the background particles and let $t \in [0,T_{\varepsilo

Figures (3)

  • Figure 1: Two particle interaction with scattering angle $\Theta$.
  • Figure 2: Description of the problem
  • Figure 3: Picture of the hyperplane $E_{v_1, v_1'}$.

Theorems & Definitions (65)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 3.1
  • Lemma 3.2
  • ...and 55 more