Homogenizations Continuum Limits and Kinetic Limits for Stochastic Models

统计力学中一个长期研究的突出问题是建立微观世界与其宏观行为之间的联系。研究者的研究涉及与稀气体演化和固体形成有关的随机模型。作为第一步，我们推导出这类随机模型宏观演化的偏微分方程。粗略地说，一个表明，经过适当的缩放，稀气体（分别为固体的边界表面）的粒子密度收敛于玻尔兹曼方程（分别为汉密尔顿-雅可比方程）的解。从概率上讲，这种收敛性是一个大数定律，它的中心极限定理为我们提供了研究微观模型的一些重要信息。固体是通过在表面发生的生长过程形成的。想象一个已经形成的原子核，周围大气中更多的物质附着在那里。附着过程是多种生长机制的一个功能，这取决于所涉及的材料、它们的温度、成分等。遵循统计力学的传统，人们研究的是一些简化的模型，而这些模型却抓住了一些基本的物理原理。这种简化的模型被证明对理解复杂的固体形成是有用的。事实证明，这些模型还可以描述其他现象，如感染细胞在组织中的扩散，杂质在流体演化中的影响等。研究者的研究涉及微观生长规律和固体表面宏观形状之间的相互作用。

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.