Mathematical foundations of plasma kinetic theory

等离子体动力学理论的数学基础

• 批准号: 1107307
• 负责人: Carlo Lancellotti
• 金额: $ 20.95万
• 依托单位: CUNY College of Staten Island
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1107307&HistoricalAwards=false
• 关键词: Mathematical; foundations; plasma; kinetic; theory

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.

本计画将着重于电浆输运理论中Balescu-Guernsey-Lenard（BGL）方程式的数学推导。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。它被提出来描述一个与长程相互作用的粒子系统的演化的无限粒子极限（例如，库仑）力。然而，没有严格的数学证明BGL方程是正确的极限。这种情况使得更难以开发用于物理情况的更通用的模型，其中BGL方程已知是不准确的（例如，聚变反应堆中的异常传输）。该项目旨在表明BGL方程和关于Vlasov极限的无碰撞等离子体波动理论之间存在着深刻的联系。具体来说，我们的目标是表明，BGL方程是简单的非线性福克-普朗克方程与长时间的粒子动力学在弗拉索夫波动场的空间均匀等离子体。研究的核心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时间、自洽的动力学过程，该随机力场是由粒子密度在平均场极限附近的波动产生的。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，关于它的现有数学知识体系非常少。该项目的主要目标是构建一个严格的数学证明，该模型提供了一个很好的近似来描述一个大的粒子集合的行为，例如在聚变等离子体中发现的那些。这一证明将验证许多基于BLG模型的理论计算和数值模拟及其在等离子体物理中的应用。更好地理解所涉及的数学对于模拟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是必不可少的。