高维周期性Lorentz气体的极限问题

The aim of this project is to study the limiting behavior of periodic Lorentz gas in high dimension. This topic stems from the goal to understand Brownian motion from microscopic particles, which inspires a lot of classical results by both mathematicians and physicists in the recent century. However this topic is far from completed and there are still a lot of problems to be studied. Our research plan consists of three parts: the large deviation principle for periodic Lorentz gas in high dimension under Boltzmann-Grad limit, the central limit theorem of periodic Lorentz gas in high dimension by taking both large time limit and Boltzmann-Grad limit, and the Boltzmann-Grad limit of periodic Lorentz gas in high dimension with an external potential.

本项目拟计划研究高维周期性Lorentz气体的极限问题。该问题的研究源于从微观粒子系统来认识布朗运动，近百年来在数学和物理领域都有许多经典的研究结果，但仍有大量问题有待研究。我们的研究计划包括三个方面，高维周期性Lorentz气体在Boltzmann-Grad极限下的大偏差问题；高维周期性Lorentz气体在长时间极限与Boltzmann-Grad极限下的中心极限定理；以及带有外势场的高维周期性Lorentz气体的Boltzmann-Grad极限及其刻画。

本项目主要研究高维周期性Lorentz气体的Boltzmann-Grad极限，对其极限行为给出更为准确的描述。我们给出了高维周期性Lorentz气体的自由程Boltzmann-Grad极限下的大偏差定理。我们考察了其Kolmogorov-Sinai熵的极限。除此之外，我们考察了高维周期性Lorentz气体的Boltzmann-Grad极限的长时间行为，给出了其超扩散过程的中心极限定理的收敛速度。

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