Collective Behavior of Stochastic Systems

研究人员的研究涉及与气体演化、团簇凝聚和碎裂以及晶体形成有关的随机和确定性模型。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，经过适当的尺度变换后，气体中的粒子密度收敛于朗道方程的解，碰撞和碎裂团簇的密度收敛于Smoluchowski方程的解，而含非均匀性（杂质）的粗糙界面由Hamilton-Jacobi偏微分方程均匀化为齐次Hamilton-Jacobi方程.与这些模型相关的许多问题还不清楚。均匀化和气体的大偏差型问题将为我们提供有关相应微观模型的有价值的信息。Smoluchowski方程的凝胶化现象仍然是一个谜，我们对相应的微观现象的理解应该有助于解决在这方面的一些开放的问题。我们的世界在不同的尺度下看起来不同！例如，流体或气体是大量分子的集合，这些分子不停地碰撞，不规则地运动，没有任何特定的目标。那么这些分子又是如何组织起来，形成大规模的流动模式的呢？大致的原因是，局部守恒定律施加的约束在微观尺度上并不直接可见。研究者的研究涉及流体，气体和晶体的微观结构和宏观行为之间的关系。由大量相互作用的粒子组成的数学模型的分析被证明是有用的，在理解我们的微观世界的复杂行为，如凝胶的形成，固体表面边界的粗糙度，在晶体中尖角的发生，.

The investigator's research concerns stochastic and deterministic models associated with the evolution of gases, coagulation and fragmentation of clusters and the formation of crystals. 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, the density of coagulating and fragmenting clusters converges to a solution of Smoluchowski Equation, and a rough interface modeled by Hamilton-Jacobi PDE with inhomogeneity (impurity) homogenizes to a homogeneous Hamilton-Jacobi equation. Many issues related to these models are not understood. Large-deviations type questions for the homogenization and gases would provide us valuable informations about the corresponding microscopic models. The phenomenon of gelation for the Smoluchowski Equation is still a mystery and our understanding of the corresponding microscopic phenomenon should help resolving some open problems in this context. 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. 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 interacting particles is proved to be useful in understanding the intricate behavior of our microscopic world such as the formation of gels, the roughness of the surface boundary of solids, the occurrence of sharp corners in crystals,....