Waves, Particle Transport and Fronts in Heterogeneous Media

异质介质中的波、粒子输运和前沿

• 批准号: 1311903
• 负责人: Leonid Ryzhik
• 金额: $ 34.78万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-10-01 至 2017-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1311903&HistoricalAwards=false
• 关键词: Waves; Particle; Transport; Fronts; Heterogeneous

AbstractNumerical simulation of the microscopic details of wave propagation in random media is still beyond reach of modern computers: a typical propagation distance may be of the order of hundreds of wavelengths and as many correlation lengths of random fluctuations. This necessitates the use of various approximate macroscopic effective models in practice. The passage from the stochastic microscopic equations modeling a particular physical system to the large-scale model is a highly non-trivial problem in itself. The goal of the first part of the project is to develop new tools and better understanding of such effective limits, especially in the regime when random media have long range correlations that lead to multiple temporal and spatial scales for various physical phenomena. The second part of the project investigates the qualitative behavior of solutions of reaction-diffusion equations, in particular, those that arise in stochastic particle systems, such as branching Brownian motion, and other Brunet-Derrida systems. Another set of problems involves reaction-diffusion equations that arise in biology, such as chemotactic models, and ecology involving population dynamics in heterogenous environments. This project carries out mathematical studies of physical systems, and wave propagation in cluttered environments and of reaction-diffusion processes. These studies are relevant to several branches of science, ranging from biomedical imaging questions to geophysics, fluid dynamics, and biology. Imaging in a cluttered environment, whether it is a human body, earth interior, or foliage, is inherently unstable because of media complexity. One objective of this project is to develop imaging methods that are less sensitive to unpredictable fluctuations of the clutter. Another area of this project concerns the mathematical description of the mixing effects of a fluid flow on chemical and biological reactions. Turbulent fluid flow plays an important role in many reaction phenomena: it may drastically enhance the rate of reaction, leading to higher efficiency, or, in some situations, extinguish the chemical process. The project will address these issues in simpler mathematical models to illuminate the mechanisms present in the full problem.

随机介质中波传播的显微镜细节的摘要数模拟仍然是现代计算机的影响力：典型的传播距离可能是数百个波长的顺序，并且随机波动的许多相关长度。这需要在实践中使用各种近似宏观有效模型。从随机显微镜方程对特定物理系统建模到大型模型的段落本身就是一个高度不平凡的问题。该项目第一部分的目的是开发新工具并更好地理解这种有效限制，尤其是在随机媒体具有长距离相关性的情况下，导致各种物理现象的多个时间和空间量表。该项目的第二部分研究了反应扩散方程的溶液的定性行为，特别是在随机粒子系统中出现的溶液，例如布朗尼分支运动和其他Brunet-Derrida系统。 另一组问题涉及生物学中出现的反应扩散方程，例如趋化模型和涉及异源环境中种群动态的生态学。该项目对物理系统进行了数学研究，并在混乱的环境和反应扩散过程中进行波传播。 这些研究与科学的几个分支有关，从生物医学成像问题到地球物理，流体动力学和生物学。在混乱的环境中进行成像，无论是人体，地球内部还是树叶，由于媒体的复杂性而来都是不稳定的。该项目的一个目的是开发对杂物不可预测的波动不太敏感的成像方法。 该项目的另一个领域涉及流体流动对化学和生物学反应的混合作用的数学描述。在许多反应现象中，湍流流动起着重要作用：它可能会大大提高反应速率，导致效率更高，或者在某些情况下，消除了化学过程。该项目将在更简单的数学模型中解决这些问题，以阐明整个问题中存在的机制。