Kinetic and Hydrodynamic Limits

[704565] Rezakhanlou统计力学中一个长期研究的突出问题是建立流体（或气体）的微观结构与其宏观行为之间的联系。也许在这方面最著名的问题是由牛顿第二定律支配的小尺度动力学推导出流体动力学和动力学方程。这个问题仍然悬而未决，但一些变体最近得到了治疗。研究者的研究涉及对模拟流体和气体的粒子系统的分析。最近，研究者与James Tarver合作，导出了一类模拟稀气体的随机粒子系统的宏观粒子密度的玻尔兹曼型方程。粗略地说，经过适当的缩放后，微观粒子密度收敛于玻尔兹曼型方程的解。研究人员还证明了收敛速度是指数级的，并导出了指数收敛速度的显式表达式。这一结果使我们能够超越玻尔兹曼方程给出的宏观描述，并对所研究的粒子系统的微观结构获得一些有价值的见解。我们的世界在不同的尺度上呈现出不同的面貌。例如，流体或气体是大量分子的集合，这些分子不停地碰撞，无规律地运动，没有任何特定的目标。那么这些分子是如何组织起来，形成大规模的流动模式的呢？大致原因是，质量、动量和能量的局部守恒定律施加了在微观尺度上无法立即看到的约束。研究者的研究涉及流体和气体的微观结构和宏观行为之间的关系。对由大量相互作用的粒子组成的数学模型的分析已证明对理解流体和气体的复杂行为是有用的。此外，相互作用的粒子系统被证明是模拟稀气体流动模式的最有效方法。

9704565 Rezakhanlou 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 recently treated. The investigator's research concerns the analysis of particle systems modeling fluids and gas. Recently the investigator in collaboration with James Tarver has derived a Boltzmann type equation for the macroscopic particle densities of a class stochastic particle systems modeling dilute gases. Roughly speaking one shows that after a suitable scaling the microscopic particle density converges to a solution of a Boltzmann type equation. The investigator also shows that the convergence is exponentially fast and derives an explicit expression for the exponential rate of convergence. This result allows us to go beyond the macroscopic description given by the Boltzmann equation and obtain some valuable insight about the microscopic structure of the particle system under 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. 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 has 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.