The convergence rates for the superdiffusion in the Boltzmann-Grad limit of the periodic Lorentz gas

One-sided convergence in the Boltzmann-Grad limit

We review various contributions on the fundamental work of Lanford deriving the Boltzmann equation from hard-sphere dynamics in the low density limit. We focus especially on the assumptions made on the initial data and on how they encode irreversibility. The impossibility to reverse time in the Boltzmann equation (expressed for instance by Boltzmann's H-theorem) is related to the lack of convergence of higher order marginals on some singular sets. Explicit counterexamples single out the microscopic sets where the initial data should converge in order to produce the Boltzmann dynamics.

Two-dimensional Lorentz process for magnetotransport: Boltzmann-Grad limit

We study a system of charged, noninteracting classical particles moving in a Poisson distribution of hard-disk scatterers in two dimensions, under the effect of a magnetic field perpendicular to the plane. We prove that, in the low-density (Boltzmann-Grad) limit, the particle distribution evolves according to a generalized linear Boltzmann equation, previously derived and solved by Bobylev et al. [4, 5, 6]. In this model, Boltzmann's chaos fails, and the kinetic equation includes non-Markovian terms. The ideas of [13] can be however adapted to prove convergence of the process with memory.

Boltzmann-Grad极限下周期洛伦兹气体的不变性原理

Invariance principle for the periodic Lorentz gas in the Boltzmann-Grad limit

在早期的工作中，我们表明，粒子位移的多维周期性洛伦兹气体，在低散射体密度的限制（玻尔兹曼-格拉德限制），满足一个中心极限定理与超扩散标度。本文将这一结果推广到一个泛函中心极限定理，即，粒子轨迹对布朗运动的弱收敛性。

Boltzmann-Grad limit of a hard sphere system in a box with diffusive boundary conditions

In this paper we present a rigorous derivation of the Boltzmann equation in a compact domain with diffuse reflection boundary conditions. We consider a system of $N$ hard spheres of diameter $ε$ in a box $Λ:= [0, 1] \times (R/Z)^2$. When a particle meets the boundary of the domain, it is instantaneously reinjected into the box with a random direction, and conserving kinetic energy. We prove that the first marginal of the process converges in the scaling $Nε^2 = 1$, $ε\rightarrow 0$ to the solution of the Boltzmann equation, with the same short time restriction of Lanford's classical theorem.

随机洛伦兹气体的半淬火不变性原理——超越玻尔兹曼-格拉德极限

Semi-Quenched Invariance Principle for the Random Lorentz Gas -- Beyond the Boltzmann-Grad Limit

通过同步耦合多条洛伦兹轨迹，将随机洛伦兹气体的退火不变性原理沿快速提取子序列提升至淬火情形，适用于R^3中随机散射体环境。

Invariance Principle for the Random Lorentz Gas -- Beyond the Boltzmann-Grad Limit

We prove the invariance principle for a \emph{random Lorentz-gas} particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius $r$, placed according to a Poisson point process of density $\varrho$, in the limit $\varrho\to\infty$, $r\to0$, $\varrho r^{2}\to1$ up to time scales of order $T=o(r^{-2}{|\log r|}^{-2})$. To our knowledge this represents the first significant progress towards solving rigorously this problem in classical nonequilibri…

每个固定维度 $d\ge 4$ 中通过双着陆根包实现的周期长时间 Boltzmann--Grad 极限

A Periodic Long-Time Boltzmann--Grad Limit in Every Fixed Dimension $d\ge 4$

本文在任意固定维度 $d\ge 4$ 中证明硬球系统的周期长时间 Boltzmann--Grad 极限，通过双着陆根包和周期首次失效构造，得到速率 $\varepsilon^{1/(400d)}$。

晶格并中 Lorentz 气体的 Boltzmann-Grad 极限

The Boltzmann-Grad Limit of the Lorentz Gas in a Union of Lattices

本文研究晶格并上 Lorentz 气体的低密度极限，借助标记构造与球面等分布验证证明其收敛为随机飞行过程，并刻画其可重构性及自由程幂律尾部。

Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations

We present a mathematical theory of dynamical fluctuations for the hard sphere gas in the Boltzmann-Grad limit. We prove that: (1) fluctuations of the empirical measure from the solution of the Boltzmann equation, scaled with the square root of the average number of particles, converge to a Gaussian process driven by the fluctuating Boltzmann equation, as predicted in [67]; (2) large deviations are exponentially small in the average number of particles and are characterized, under regularity assumptions, by a large deviation functional as previously obtained in [61] for dynamics with stochasti…