Large Deviation, Kinetic Limit and Gelation

建立微观世界与宏观行为之间的联系是统计力学中一个长期研究的突出问题。研究人员的研究涉及与气体演化、凝胶和晶体的形成以及具有随机势的哈密顿系统相关的随机和确定性模型。作为第一步，我们导出了这类随机模型的宏观演化的偏微分方程。粗略地说，经过适当的标度，掠过碰撞气体中的粒子密度收敛到Landau方程的解，由含杂质的Hamilton-Jacobi PDE模拟的粗糙界面均匀化为齐次的Hamilton-Jacobi方程，具有随机势的粒子链由有效势控制。与这些模型相关的许多问题还没有完全理解。关于均匀化的大偏差型问题和关于气体的中心极限定理型结果将为我们提供关于相应微观模型的有价值的信息。此外，凝聚现象的简化模型应该有助于我们发现支配凝胶和溶胶之间神秘相互作用的规律。我们的世界在不同的尺度上呈现出不同的样子！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？大致原因是，当地的保护法施加了微观层面上看不到的限制。作为另一个例子，考虑一种固体，该固体从环境大气中粘着更多的材料。附着的过程是各种生长机制的函数，取决于所涉及的材料、温度、成分等。遵循统计力学的传统，人们研究简化的模型，但它捕捉了一些基本的物理。研究人员的研究涉及微观结构与流体、气体和晶体的宏观行为之间的关系。对由大量组分组成的数学模型的分析被证明有助于理解我们微观世界的复杂行为，如晶体的形成、血液中凝块的出现、固体表面边界的粗糙以及流体中冲击的发生。

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 and deterministic models associated with the evolution of gases, the formation of gels and crystals, and Hamiltonian systems with random potentials. As the first step, one derives a partial differential equation for the macroscopic evolution of such stochastic models. Roughly, after a suitable scaling, the density of particles in a gas with grazing collisions converges to a solution of Landau Equation, a rough interface modeled by Hamilton-Jacobi PDE with impurity homogenizes to a homogeneous Hamilton-Jacobi equation, and a chain of particles with random potential are governed by an effective potential. Many issues related to these models are not fully understood. Large-deviations type questions for the homogenization and a central limit theorem type result for gases would provide us with valuable information about the corresponding microscopic models. Also, simplified models for coagulation phenomenon should help us to discover the rules governing the mysterious interaction between gels and sols.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 impose constraints not immediately visible on the microscopic scale. As an another example, consider a solid 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.The investigator's research concerns the relationship between the microscopic structure and the macroscopic behavior of fluids, gases and crystals. The analysis of the mathematical models consisting of a large number of components is proved to be useful in understanding the intricate behavior of our microscopic world such as the formation of crystals, the emergence of clots in blood, the roughness of the surface boundary of solids and occurrence of shocks in fluids.