Homogenizations Continuum Limits and Kinetic Limits for Stochastic Models

建立微观世界与宏观行为之间的联系是统计力学中一个突出而长期研究的问题。研究者的研究涉及与稀气体演化和固体形成相关的随机模型。作为第一步，推导出这样的随机模型的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，粒子密度的稀释气体（相应地，固体的边界表面）收敛到玻尔兹曼方程（相应地，哈密尔顿-雅可比方程）的解决方案。从概率上讲，这种收敛是一个大数定律，其相应的中心极限定理为我们提供了一些关于所研究的微观模型的重要信息。 固体通过在表面发生的生长过程形成。想象一下，一个已经形成的原子核，周围大气中的其他物质粘在它上面。附着的过程是取决于所涉及的材料，它们的温度，成分等的各种各样的生长机制的函数。这种简化的模型被证明是有用的，在理解复杂的固体形成。事实证明，这些模型描述了其他现象，如组织中感染细胞的传播，流体演化中杂质的影响等。研究人员的研究涉及微观生长规则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.