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Wave transport in low-density matter, Siegel theta functions, and homogeneous flows

低密度物质中的波传输、西格尔 theta 函数和均匀流

基本信息

批准号：
EP/S024948/1
负责人：
Jens Marklof
金额：
$ 81.52万
依托单位：
University of Bristol
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2019
资助国家：
英国
起止时间：
2019 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FS024948%2F1
关键词：
Wave transport low density matter

项目摘要

This proposal addresses the fundamental challenge of understanding wave transport in a given medium. The subject is extremely broad, ranging from experimental science to theoretical modelling. Our focus will be on the rigorous mathematical derivation of transport equations from the underlying fundamental laws of physics, and to thus describe effects on scales which are several orders of magnitude above the length scale given by the fine structure of the medium. The exciting aspect of the proposed research is that some of the transport processes we seek to derive are new and will expose subtle corrections to the classical linear Boltzmann equation. The findings of this project will thus not only be of fundamental interest in mathematics and mathematical physics, but also in applied areas where the linear Boltzmann equation serves as a central model; examples include radiative transfer, neutron scattering and semiconductor physics. The tools we employ build on recently developed techniques on the geometric regularisation of theta functions, which in turn uses the dynamics of group actions on homogeneous spaces. The further development of these deep and sophisticated methods will form a major part of this project, and will have independent applications in long-standing questions on the distribution of quadratic forms (i.e. quantitative versions of the Oppenheim conjecture for values in shrinking intervals) and the value distribution of theta functions.

这一建议解决了理解给定介质中的波传输的根本挑战。这门学科非常宽泛，从实验科学到理论建模。我们的重点是从基本的物理基本定律对输运方程进行严格的数学推导，从而描述在介质精细结构所给出的长度尺度上几个数量级以上的尺度上的影响。这项拟议的研究令人兴奋的方面是，我们试图推导出的一些输运过程是新的，将揭示对经典线性玻尔兹曼方程的微妙修正。因此，该项目的成果不仅将在数学和数学物理方面引起基本兴趣，而且还将在以线性玻尔兹曼方程为中心模型的应用领域产生兴趣；例如辐射传输、中子散射和半导体物理。我们使用的工具建立在最近开发的关于theta函数的几何正则化的技术之上，而这反过来又使用了齐次空间上的群体行动的动力学。这些深入而复杂的方法的进一步发展将成为这个项目的主要部分，并将在二次型分布(即收缩区间值的Oppenheim猜想的量化版本)和theta函数的值分布的长期问题中有独立的应用。