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.
The 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.
