Mathematical Sciences: The Wigner-Poisson and Related Systems

数学科学：维格纳-泊松及相关系统

• 批准号: 9403458
• 负责人: Paul Zweifel
• 金额: $ 9万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403458&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Wigner; Poisson; Related

9403458 Zweifel The Wigner Poisson System of equations describes electron plasmas in a regime where quantum effects are important, for example semi- conductor devices including field-effect transistors, Josephson junctions, hetero-junctions, quantum "dots" and the like. A self- consistent field approach is employed leading to pseudo-differential equations which in the classical limit tend to the ordinary Vlasov- Poisson system. The problem we propose to study includes existence and uniqueness of solutions, their regularity and asymptotic properties and developement of numerical methods appropriate to the treatment of practical devices, as well as studies of the classical limiting procedure. The studies to be carried out under this proposal are intended to lead to an understanding of methods applicable to many practical systems of interest in physics and engineering, especially systems of small size, because that is where quantum effects become important. Another important facet of our work will be to develop methods for obtaining quantum corrections to systems which are quasi-classical in nature, since such calculations will provide great economy in computer time and cost, vis-a-vis a full quantum description. Such calculations have been carried out by some researchers, but without any proof or even reasonable grounds to believe that such a procedure is mathematically valid.

小行星9403458 Wigner Poisson方程系统描述了量子效应很重要的情况下的电子等离子体，例如包括场效应晶体管、约瑟夫森结、异质结、量子“点”等的半导体器件。采用自洽场方法，导出了在经典极限下趋于普通Vlasov-Poisson系统的伪微分方程。我们建议研究的问题包括解决方案的存在性和唯一性，其正则性和渐近性质和发展的数值方法适当的实际设备的治疗，以及经典的限制程序的研究。 根据这一建议进行的研究旨在了解适用于物理学和工程学中感兴趣的许多实际系统的方法，特别是小尺寸系统，因为这是量子效应变得重要的地方。我们工作的另一个重要方面将是开发获得准经典系统的量子修正的方法，因为这样的计算将在计算机时间和成本方面提供极大的经济性，相对于完整的量子描述。一些研究人员进行了这样的计算，但没有任何证据，甚至没有合理的理由相信这样的程序在数学上是有效的。