Mathematical Sciences: Exact Results in Statistical Mechanics

在这个项目中，首席研究员将研究当今统计力学中一个重要的基本问题，即建立微观现象和宏观现象之间的联系。更具体地说，在量子力学的背景下，他将试图理解描述许多粒子相互作用的微观定律与宏观可观察到的现象之间的关系，例如表征现象的总体特性的温度。这里的部分困难在于物理学家无法就量子力学形式论的微观定律达成一致。首席研究员将尝试建立这样的规律，然后以严格的方式检查其现象学结果。物理学家和应用数学家称之为咖啡的“宏观”或体积特性的是一种容易描述和观察到的现象，比如你早晨一杯咖啡的温度。也就是说，它是一种可测量的量，可以根据某种常规尺度来描述液体的热或冷。然而，温度实际上是由杯子内不断运动的许多分子和原子的集体行为决定的，这些分子和原子受相互作用的“微观”定律支配。在这个提议中，首席研究员将在量子力学的背景下研究微观和宏观效应之间的关系，量子力学是物质的终极理论。

In this project the principal investigator will look into a fundamental problem of importance in statistical mechanics today, namely establishing connections between microscopic phenomena and macroscopic phenomena. More specifically, he will try to understand the relationship between microscopic laws that describe the interactions of many particles and macroscopic observables such as temperature that characterize the bulk properties of a phenomenon, in the context of quantum mechanics. Part of the difficulty here lies in the fact that physicists cannot agree about what the microscopic laws are for the quantum mechanical formalism. The principal investigator will attempt to establish such laws and then examine their phenomenological consequences in a rigorous fashion. An easily describable and observable phenomenon such as the temperature of your morning cup of coffee is what physicists and applied mathematicians call a "macroscopic" or bulk property of the coffee, that is, it is a measurable quantity that describes how hot or cold the liquid is, based upon some conventional scale. The temperature however is actually governed by the collective behavior of the many molecules and atoms that are in ceaseless motion inside of the cup and that are governed by the "microscopic" laws of interaction. In this proposal the principal investigator will examine the relationship between microscopic and macroscopic effects in the context of quantum mechanics, the ultimate theory of matter.