Mathematical Sciences: Infinite Particle Systems and Hydrodynamics

首席研究员正致力于推导随机伊辛模型的宏观方程。在不同相共存的低温下对这一问题的研究表明，不同相的液滴按照某种类型的“平均曲率运动”PDE来演化。对于该模型或其适当的变体，如简单的变速排除或Ginzburg-Landau模型，都可以提出相关的问题。另一个令人感兴趣的问题是非对称简单排除的守恒定律的推导。在这个模型中分析标记粒子的渐近行为也是很有意义的。统计力学中的一项重要任务是从其分子结构推导出流体的演化。在分子水平上，流体是由大量分子按照物理定律运动组成的。在显微镜下，分子不断地碰撞，并以类似于随机的方式有点不规律地运动。由于质量守恒定律、动量守恒定律和熵守恒定律，分子能够以这样一种方式自我组织，形成大规模的流动模式。将研究流体的各种漫画模型，以严格的数学方式从其微观结构推导出它们的宏观演化。

The principal investigator is working on the derivation of a macroscopic equation for the stochastic Ising model. Studies of the problem at low temperatures where different phases coexist indicate that the droplets of different phases evolve according to some type of "motion by mean curvature" PDE. Related problems can be proposed for either this model or its appropriate variations such as simple exclusions with speed change or the Ginzburg-Landau model. Another problem of interest is the derivation of conservation laws for asymmetric simple exclusions with speed changes. Analyzing the asymptotic behavior of a tagged particle in this model is also of great interest. An important task in statistical mechanics is to derive the evolution of a fluid from its molecular structure. At the molecular level, a fluid consists of a large number of molecules moving according to physical laws. Microscopically, molecules collide incessantly and move somewhat erratically in a random like fashion. Due to conservation laws of mass, momentum and entropy, molecules manage to organize themselves in such a way as to form a flow pattern on a large scale. Various Caricature models of a fluid will be studied to derive their macroscopic evolution from their microscopic structure in a mathematically rigorous fashion.