Homogenizations Continuum Limits and Kinetic Limits for Stochastic Models

随机模型的均质化连续体极限和动力学极限

• 批准号: 0072666
• 负责人: Fraydoun Rezakhanlou
• 金额: $ 9万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072666&HistoricalAwards=false
• 关键词: Homogenizations; Continuum; Limits; Kinetic; Stochastic

An outstanding and long studied problem in statistical mechanics is to establish the connection between the microscopic world and its macroscopic behavior. The investigator's research concerns stochastic models associated with the evolution of dilute gases and the formation of solids. As the first step, one derives a partial differential equation for the macroscopic evolution of such stochastic models. Roughly speaking, one shows that after a suitable scaling, the particle density of a dilute gas (respectively, the boundary surface of a solid) converges to a solution of the Boltzmann equation (respectively, Hamilton-Jacobi equation). Probabilistically, such a convergence is a law of large numbers and its corresponding central limit theorem provides us with some vital information about the microscopic model under the study. Solids form through growth processes which take place at the surface. Imagine an already formed nucleus to which further material sticks from the ambient atmosphere. The process of the attachment is a function of a huge variety of growth mechanisms depending on the materials involved, their temperature, composition, etc. Following the tradition of statistical mechanics, one studies simplified models which nevertheless captures some of the essential physics. Such simplified models are proved to be useful in understanding the intricate formation of solids. It turns out that these models describe other phenomena such as the spread of infected cells in a tissue, the effect of impurity in the evolution of a fluid, etc. The investigator's research concerns the interplay between the microscopic growth rules and the macroscopic shape of the surface of a solid.

统计力学中一个突出且长期研究的问题是建立微观世界与其宏观行为之间的联系。研究人员的研究涉及与稀气体演化和固体形成相关的随机模型。第一步，推导此类随机模型的宏观演化的偏微分方程。粗略地说，表明经过适当的缩放后，稀气体（分别为固体的边界面）的粒子密度收敛于玻尔兹曼方程（分别为哈密尔顿-雅可比方程）的解。从概率上讲，这种收敛是大数定律，其相应的中心极限定理为我们提供了有关所研究的微观模型的一些重要信息。 固体通过在表面发生的生长过程形成。想象一个已经形成的核，周围大气中的更多物质粘附在该核上。附着过程是多种生长机制的函数，具体取决于所涉及的材料、温度、成分等。遵循统计力学的传统，人们研究了简化的模型，但仍然捕获了一些基本的物理原理。事实证明，这种简化模型有助于理解固体的复杂形成。事实证明，这些模型描述了其他现象，例如组织中受感染细胞的传播、流体演化中杂质的影响等。研究人员的研究涉及固体表面的微观生长规则和宏观形状之间的相互作用。