Mathematical Sciences: Interacting Particle Systems and Hydrodynamics

数学科学：相互作用的粒子系统和流体动力学

• 批准号: 9424270
• 负责人: Fraydoun Rezakhanlou
• 金额: $ 4万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9424270&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Interacting; Particle; Systems

9424270 Rezakhanlou Abstract An outstanding and long studied problem in statistical mechanics is to establish the connection between the microscopic structure of a fluid (or a gas) and its macroscopic behavior. Perhaps the most celebrated problem in this context is the derivation of the Hydrodynamic and kinetic equations from small scale dynamics governed by the Newton's second law. This problem is still wide open but some variants have been reaently treated. The investigator's research concerns the analysis of particle systems simulating a fluid or a dilute gas. Recently the investigator has derived the Boltzmann equation for the macroscopic particle densities for a class of one dimensional lattice gas models. Roughly speaking one shows that after a suitable scaling the microscopic particle density converges to a solution of the Boltzmann equation. The convergence is expected to be exponentially fast and the exponential rate can be expressed by a variational formula. Our world appears differently at different scales! For example a fluid or a gas is a collection of an enormous number of molecules that collide incessantly and move erratically without any particular aim. How do these molecules then manage to organize themselves in such a way as to form a flow pattern on a large scale? Roughly the reason is that the local conservation laws of mass, momentum and energy impose constraints not immediately visible on the microscopic scale. The investigator's research concerns the relationship between the microscopic structure and the macroscopic behavior of fluids and gases. The analysis of the mathematical models consisting of a large number of interacting particles is proved to be useful in understanding the intricate behavior of fluids and gases. Moreover, interacting particle systems turn out to be the most efficient way of simulating the flow patterns of dilute gases.

9424270 Rezakhanlou摘要统计力学中一个长期研究的突出问题是建立流体(或气体)的微观结构与其宏观行为之间的联系。在这种情况下，最著名的问题可能是由牛顿第二定律支配的小尺度动力学推导出流体动力学和动力学方程。这个问题仍然悬而未决，但一些变种已经得到了真正的解决。这位研究人员的研究涉及模拟流体或稀薄气体的粒子系统的分析。最近，研究人员导出了一类一维晶格气体模型的宏观粒子密度的Boltzmann方程。粗略地说，在适当的尺度下，微观粒子密度收敛于玻耳兹曼方程的解。期望收敛速度是指数级的，并且指数速度可以用一个变分公式来表示。我们的世界在不同的尺度上呈现出不同的面貌！例如，流体或气体是大量分子的集合，这些分子不断碰撞并无特定目标地不规则运动。那么这些分子是如何以这样一种方式进行自我组织以形成大规模流动模式的？大致的原因是，当地的质量、动量和能量守恒定律施加了在微观尺度上看不到的约束。研究人员的研究涉及流体和气体的微观结构和宏观行为之间的关系。对由大量相互作用的粒子组成的数学模型的分析被证明有助于理解流体和气体的复杂行为。此外，相互作用的粒子系统被证明是模拟稀薄气体流动模式的最有效方法。