Hamiltonian and Dissipative Structures for Reduced Plasma Kinetic Models

该奖项支持简化等离子体模型的理论发展，该模型允许对带电粒子和电磁场进行精确和同时的处理。对等离子体物理现象的描述，无论是在实验室还是在太空中观察到的，都必须考虑带电等离子体粒子轨道与控制其动力学的电磁场之间复杂的相互作用。然而，在现实的等离子体条件下，潜在的等离子体方程往往太复杂而无法解决，即使有最强大的计算机的帮助。采用简化等离子体模型可以简化等离子体物理描述的复杂性，这些模型只有在原始等离子体方程中保留其精确守恒性质时才能忠实于精确的等离子体动力学。该项目的一个主要成果是推导出具有精确守恒特性的各种简化等离子体模型的新理论结构，这些模型允许在一般等离子体条件下同时处理粒子和场。在简化等离子体模型中同时处理粒子和场可以显著提高为这些简化等离子体模型开发的新数值算法的功率和速度。反过来，这些新的数值算法将使实验室和空间磁化等离子体在更长的时间尺度和更现实的条件下的复杂动力学研究成为可能。在过去的十年中，等离子体动力学模型的哈密顿和耗散支架结构的存在导致了新的结构保持计算算法的出现。在本项目中进行的这些结构保持算法的一个重要的新应用涉及到简化等离子体陀螺动力学模型的哈密顿和耗散支架结构的发展。对于一组给定的陀螺动力学等离子体方程，其哈密顿支架结构的推导在理论物理学中提出了独特的挑战，是本项目的主要成果。根据场而不是势来制定回旋动力学等离子体模型，将开辟空间和天体物理等离子体物理学研究的新前沿。陀螺动力学哈密顿托架结构也允许在一般磁几何下推导陀螺动力学朗道碰撞算符的耗散托架结构。通过构造，耗散陀螺动力学支架结构将具有精确的守恒性质，这是研究长时间尺度上无碰撞（哈密顿）和碰撞（耗散）等离子体演化之间协同作用的关键要求。因此，与简化等离子体回旋动力学模型相关的先进哈密顿和耗散结构保持算法的发展将为研究与实验室、空间和天体物理磁化等离子体的非线性湍流演化相关的更广泛的研究问题提供一套强大的理论和计算工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award supports theoretical development of reduced plasma models that allow for accurate and simultaneous treatment of charged particles and electromagnetic fields. The description of plasma physics phenomena, whether observed in a laboratory or in space, must consider the complex interactions between the orbits of charged plasma particles and the electromagnetic fields that control their dynamics. In realistic plasma conditions, however, the underlying plasma equations are often too complex to be solved, even with the help of the most powerful computers available. The complexity of the plasma physics description can be simplified by the adoption of reduced plasma models, which are faithful to the exact plasma dynamics only if their exact conservation properties are preserved from the original plasma equations. A major outcome of this project is the derivation of new theoretical structures for a variety of reduced plasma models with exact conservation properties that allow for the simultaneous treatment of particles and fields under general plasma conditions. The simultaneous treatment of particles and fields within a reduced plasma model can yield significant increases in the power and speed of new numerical algorithms developed for these reduced plasma models. These new numerical algorithms, in turn, will enable the study of the complex dynamics of laboratory and space magnetized plasmas over longer time scales and under more realistic conditions. The existence of Hamiltonian and dissipative bracket structures for plasma kinetic models have led to the emergence of novel structure-preserving computational algorithms in the past ten years. An important new application of these structure-preserving algorithms carried out in this project is concerned with the development of Hamiltonian and dissipative bracket structures for reduced plasma gyrokinetic models. The derivation of a Hamiltonian bracket structure for a given set of gyrokinetic plasma equations, which presents unique challenges in theoretical physics, is a major outcome of this project. The formulation of gyrokinetic plasma models in terms of fields, instead of potentials, will open new frontiers of research in space and astrophysical plasma physics. The gyrokinetic Hamiltonian bracket structure will also allow for the derivation of the dissipative bracket structure of the gyrokinetic Landau collision operator under general magnetic geometries. By construction, the dissipative gyrokinetic bracket structure will have exact conservation properties, which is a crucial requirement to investigate the synergy between collisionless (Hamiltonian) and collisional (dissipative) plasma evolutions over long time scales. The development of advanced Hamiltonian and dissipative structure-preserving algorithms associated with reduced plasma gyrokinetic models will thus provide a set of powerful theoretical and computational tools for investigations of a wider range of research problems associated with the nonlinear turbulent evolution of laboratory, space, and astrophysical magnetized plasmas.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.