Macroscopic Properties of Quantum Mechanical Systems

将考虑具有大量量子粒子的系统的三个方面：非平衡输运性质，玻色子粒子在低温下的行为，以及离散拉普拉斯的性质。我们将研究在不同温度和化学势下几个宏观延伸金属之间的热接触或隧道结中发生的输运过程。目标是更好地理解昂萨格关系，该关系声称电流与热力学力的依赖具有一定的对称性。相互作用玻色子的系统将在费曼-卡茨表示中考虑。我们将研究以下三种玻色-爱因斯坦凝聚方法之间的联系：费曼的无限循环概念；产生具有线性色散关系的激励气体的Bogolubov近似；以及彭罗斯和昂萨格的非对角长程序的概念。最后，我们将研究任意有限立方晶格子集上离散拉普拉斯算子的最低特征值和。它表示非相互作用电子的基态能量。目标是展示边界的效果。量子世界是一个奇怪的世界，许多物理实验都违背了直觉。量子力学系统的一般知识和实验研究的能力在很大程度上受益于理论研究。数学物理研究的中心是这样一个事实，即宏观量是由大量微观粒子的贡献的平均值给出的，这些平均值可以用统计力学的工具计算出来。这个项目的一个方面是发展涉及很多粒子的系统的数学描述。另一个方面是玻色子粒子系统的研究，如氦，其低温行为最近引起了人们的广泛关注。我们将考虑几何方法，其中量子粒子由具有任意圈数的时空轨迹表示。目标是更深入地了解相互作用粒子系统中的玻色-爱因斯坦凝聚。

Three aspects of systems with large numbers of quantum particles will be considered: Non-equilibrium transport properties, the behavior of bosonic particles at low temperatures, and properties of the discrete Laplacian. We will study transport processes that take place in thermal contacts or tunneling junctions between several macroscopically extended metals at different temperatures and chemical potentials. A goal is to better understand the Onsager relations, which claim a certain symmetry in the dependence of currents from thermodynamic forces. Systems of interacting bosons will be considered in the Feynman-Kac representation. We will investigate the links between the following three approaches to Bose-Einstein condensation: Feynman's notion of infinite cycles; Bogolubov's approximations that result in a gas of excitations with a linear dispersion relation; and Penrose and Onsager's notion of off-diagonal long-range order. Finally, we will study the sum of lowest eigenvalues of the discrete Laplacian on arbitrary finite subsets of the cubic lattice. It represents the ground state energy of non-interacting electrons. The goal is to exhibit the effects of the boundary.brbrThe quantum world is a strange one, and many physical experiments defy intuition. General knowledge of quantum mechanical systems and the ability to study them experimentally considerably benefit from theoretical investigations. This study in mathematical physics is centered on the fact that macroscopic quantities are given by averages over the contribution of a huge number of microscopic particles, and these averages can be computed with the tools of statistical mechanics. One aspect of this project is the development of the mathematical description of systems involving very many particles. Another aspect is the study of systems of bosonic particles, such as helium, whose low-temperature behavior has recently attracted much attention. We will consider geometric approaches where quantum particles are represented by space-time trajectories with arbitrary winding numbers. The goal is a deeper understanding of the Bose-Einstein condensation in systems of interacting particles.