Mathematical Sciences: Some Problems in Statistical Mechanics

8900244 COLON这个项目与统计力学中的一些问题有关。统计力学的基本目的是从粒子之间的微观相互作用的知识中获得多粒子系统的大尺度热力学行为。热力学行为被描述为几个参数的函数，如温度和命运。首席研究员将研究高温下具有短程相互作用的经典气体。他计划证明，在足够高的温度下，没有相变。主要任务是研究高密度系统，并证明它是近似的高斯系统。他还将研究具有长程库仑相互作用的系统的基态能量。这些系统特别重要，因为电磁库仑力在自然界中无处不在。对于胶体背景中的带电玻色子系统，他想要证明，在高密度下，每个粒子的基态能量的热力学极限由Foldy公式精确给出。这将表明，在超流和超导的启发式理论中广泛使用的Bogoliubov ansatz在某种情况下是有效的。

8900244 Conlon This project is concerned with some problems in statistical mechanics. The basic aim of statistical mechanics is to obtain the large scale thermodynamic behavior of a many particle system from a knowledge of the microscopic interaction between the particles. The thermodynamic behavior is described as a function of a few parameters such as temperature and destiny. The principal investigator will study a classical gas with short range interaction at high temperature. He plans to show that at a sufficiently high temperature there is no phase transition. The main task is to study the high density system and show that it is approximately gaussian. He will also study ground state energies of systems with long range Coulomb interaction. These systems are particularly important since the electromagnetic Coulomb force is ubiquitous in nature. For a system of charged Bosons in a jellium background, he wants to show that the thermodynamic limit of the ground state energy per particle is, at high densities, given precisely by a formula of Foldy. This would show that the Bogoliubov ansatz, which is widely used in heuristic theories of superfluidity and superconductivity, is valid in a certain case.