We prove an invariance principle for a 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 this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (1970), Spohn (1978) and Boldrighini-Bunimovich-Sinai (1983). The novelty is that the diffusive scaling of particle trajectory and the kinetic (Boltzmann-Grad) limit are taken simulataneously. The main ingredients are a coupling of the mechanical trajectory with the Markovian random flight process, and probabilistic and geometric controls on the efficiency of this coupling.

中文翻译：

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随机洛伦兹气体的不变性原理——超越 Boltzmann-Grad 极限

我们证明了 3 维随机洛伦兹气体粒子在 Boltzmann-Grad 极限和同时扩散标度下的不变性原理。也就是说，对于一个点状粒子在半径为 $r$ 的无限质量、硬核、球形散射体之间移动的轨迹，根据密度 $\varrho$ 的泊松点过程放置，在极限 $\varrho \to\infty$, $r\to0$, $\varrho r^{2}\to1$ 直到时间尺度 $T=o(r^{-2}{|\log r|}^{- 2})$。据我们所知，这是自 Gallavotti (1970)、Spohn (1978) 和 Boldrighini-Bunimovich-Sinai (1983) 的开创性工作以来在经典非平衡统计物理学中解决这个问题的第一个重大进展。新颖之处在于同时采用粒子轨迹的扩散缩放和动力学（Boltzmann-Grad）极限。