Scaling Limits for Microscopic Models

统计力学中一个基本的和长期研究的问题是建立微观世界和宏观行为之间的联系。该研究员的研究涉及与稀气体演化、凝聚-碎裂现象和固体形成有关的随机模型。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。在稀释气体的情况下，一个表明，经过适当的缩放，粒子密度收敛到玻尔兹曼方程的解。在凝聚-破碎现象的情况下，重新缩放的微观颗粒密度收敛于Smoluchowski方程的解。作为晶体的模型，可以考虑具有粘性的非齐次Hamilton-Jacobi方程。经过适当的尺度变换后，解收敛到齐次Hamilton-Jacobi方程的解。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和大偏差原理为我们提供了一些关于所研究的微观模型的重要信息。 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。然而，这些分子设法以这样一种方式组织自己，从而形成大规模的流动模式。研究者的研究涉及流体，气体和固体的微观结构和宏观行为之间的关系。分析由大量相互作用的粒子组成的数学模型被证明是有用的，在理解流体和气体的复杂行为。此外，相互作用粒子系统被证明是模拟稀气体流动模式的最有效方法。

A fundamental 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, coagulation-fragmentation phenomenon and the formation of solids. As the first step, one derives a partial differential equation for the macroscopic evolution of such stochastic models. In the case of a dilute gas, one shows that after a suitable scaling, the particle density converges to a solution of the Boltzmann equation. In the case of the coagulation-fragmentation phenomenon, the rescaled microscopic particle density converges to a solution of the Smoluchowski equation. As a model of a crystal, one may consider an inhomogeneous Hamilton-Jacobi equation with viscosity. After a suitable scaling, the solutions converge to solutions of homogeneous Hamilton-Jacobi equation. Probabilistically, such a convergence is a law of large numbers and its corresponding central limit theorem and large deviation principle provide us with some vital information about the microscopic model under the study. 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. However these molecules manage to organize themselves in such a way as to form a flow pattern on a large scale. The investigator's research concerns the relationship between the microscopic structure and the macroscopic behavior of fluids, gases and solids. 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 fluids and gases. Moreover, interacting particle systems turn out to be the most efficient way of simulating the flow patterns of dilute gases.