Mathematical Sciences: Interacting Particle Systems and Hydrodynamics

9424270 Rezakhanlou摘要 在统计力学中，一个突出的和长期研究的问题是建立流体（或气体）的微观结构与其宏观行为之间的联系。也许最著名的问题在这方面是推导的流体动力学和动力学方程的小规模动态牛顿第二定律。这个问题仍然是开放的，但一些变种已被reently治疗。研究人员的研究涉及模拟流体或稀释气体的粒子系统的分析。最近，研究者导出了一类一维格子气模型的宏观粒子密度的Boltzmann方程。粗略地说，一个表明，经过适当的缩放后，微观粒子密度收敛到玻尔兹曼方程的解。该方法的收敛速度是指数级的，且指数级的收敛速度可用变分公式表示。 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，质量、动量和能量的局部守恒定律施加了在微观尺度上不能立即看到的约束。研究者的研究涉及流体和气体的微观结构与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。

9424270 Rezakhanlou Abstract An outstanding and long studied problem in statistical mechanics is to establish the connection between the microscopic structure of a fluid (or a gas) and its macroscopic behavior. Perhaps the most celebrated problem in this context is the derivation of the Hydrodynamic and kinetic equations from small scale dynamics governed by the Newton's second law. This problem is still wide open but some variants have been reaently treated. The investigator's research concerns the analysis of particle systems simulating a fluid or a dilute gas. Recently the investigator has derived the Boltzmann equation for the macroscopic particle densities for a class of one dimensional lattice gas models. Roughly speaking one shows that after a suitable scaling the microscopic particle density converges to a solution of the Boltzmann equation. The convergence is expected to be exponentially fast and the exponential rate can be expressed by a variational formula. 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 of mass, momentum and energy 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 and gases. 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.