Mathematical Sciences: Exact Results in Statistical Mechanics

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.