Mathematical foundations of plasma kinetic theory

本项目将集中于等离子体输运理论的巴列斯库-根西-勒纳德(BGL)方程的数学推导。这个方程通常被认为是经典库仑等离子体最基本的动力学碰撞模型。它被建议用来描述与长程(例如库仑)力相互作用的粒子系统的演化的无限粒子极限。然而，没有严格的数学证明来证明BGL方程是正确的极限。这种情况使得为物理情况开发更一般的模型变得困难得多，在这种情况下，已知BGL方程不准确(例如，聚变反应堆中的异常传输)。这个项目的目的是证明BGL方程和无碰撞等离子体中关于弗拉索夫极限的涨落理论之间存在着深刻的联系。具体地说，目标是证明BGL方程是空间均匀等离子体的Vlasov涨落场中与长时间粒子动力学相关的非线性Fokker-Planck方程。这项研究的中心将是一个新的随机过程，它描述了粒子在随机力场中运动的长期自洽动力学，该过程是由粒子密度在平均场极限附近的涨落所产生的。Balescu-Lenard-Guernsey方程是等离子体和引力系统理论中的关键方程，它有许多应用，特别是在磁约束聚变反应堆的数学建模中。然而，现有的关于它的数学知识非常少。该项目的主要目标是构建一个严格的数学证明，证明该模型为描述大量粒子的行为提供了一个很好的近似，例如聚变等离子体中发现的粒子。这样的证明将验证许多基于BLG模型的理论计算和数值模拟，以及它们在等离子体物理中的应用。对于聚变反应堆中非常热的等离子体中出现的更复杂的物理现象进行建模，更好地理解所涉及的数学将是必不可少的。

This project will focus on the mathematical derivation of the Balescu-Guernsey-Lenard (BGL) equation of plasma transport theory. This equation is commonly regarded as the most fundamental kinetic collisional model for classic Coulomb plasmas. It was proposed to describe the infinite-particle limit for the evolution of a system of particles that interact with long-range (e.g., Coulomb) forces. However, there is no rigorous mathematical proof that the BGL equation is the correct limit. This situation has made it much harder to develop more general models for the physical situations, where the BGL equation is known not to be accurate (e.g., anomalous transport in fusion reactors). This project will aim to show that there is a deep connection between the BGL equation and the theory of fluctuations in collisionless plasmas about the Vlasov limit. Specifically, the goal is to show that the BGL equation is simply the non-linear Fokker-Planck equation associated with long-time particle dynamics in the Vlasov fluctuation field of a spatially homogeneous plasma. The centerpiece of the study will be a new stochastic process which describes, loosely speaking, long-time, self-consistent dynamics of particles traveling in a random force field generated by the fluctuations of the particles' density about the mean-field limit.The Balescu-Lenard-Guernsey equation is pivotal in the theory of plasmas and gravitating systems; it has many applications, especially in the mathematical modeling of magnetically-confined fusion reactors. However, the existing body of mathematical knowledge about it is very small. The chief goal of the project is to construct a rigorous mathematical proof that this model provides a good approximation for describing the behavior of a large assembly of particles, such as those found in fusion plasmas. Such a proof will validate many theoretical calculations and numerical simulations based on the BLG model, as well as their applications in plasma physics. Having a better understanding of the mathematics involved will be indispensable for modeling more complicated physical phenomena that arise in very hot plasmas in a fusion reactor.