Context: In our description of nature with quantum mechanics we encounter two sorts of particles, bosons and fermions. From the Pauli exclusion principle we know that already two non-interacting fermions (e.g. electrons, protons, certain atoms) can never occupy the same quantum state (they can for example never be at the same point in space) — we say they have a statistical repulsion. Bosons (e.g. photons, certain atoms) in contrast have a statistical attraction and already the ideal (non-interacting) Bose gas shows an interesting phenomenon called the Bose—Einstein condensation (BEC) phase transition: Below a certain critical temperature a macroscopic fraction of all particles in the gas starts to behave in exactly the same way. The BEC phase transition is a purely quantum mechanical effect and has no classical counterpart.
Bose gases, and more generally quantum gases, play a prominent role in modern physics because they allow for the simulation of a large variety of complex quantum many-particle systems with room-size experimental set-ups. With these experiments it is possible to obtain precise information about the physics of systems consisting of a huge number of degrees of freedom (e.g. particles), whose properties are not computable even with the largest supercomputers on earth. The persisting goal of these studies is to obtain a better understanding of how the microscopic constituents of such systems determine their macroscopic properties as e.g. their electrical conductivity, and to shed light on the physical origin of such important effects as e.g. high-temperature superconductivity. Mathematical physicists contribute in this endeavour by proving the existence of prominent physical effects as generally as possible starting from their fundamental quantum mechanical description, and by rigorously deriving effective equations used by physicists to describe them. Apart from their mathematical content, such proofs usually also allow us to learn more about the physics of the problem.
The overall objective of the project “Dilute Bose Gases at Positive Temperature” was to develop new mathematical tools to study thermodynamic properties of dilute Bose gases at positive temperature (in contrast to zero temperature) as well as the dynamics of approximate thermodynamic equilibrium states after external electric and/or magnetic fields have been changed (e.g. a trapping potential has been switched off). The project, which has been intended for a period of two years, has been ended ahead of schedule after nine months in favour of a lecturer position (4 years) at the Institute of Mathematics of the University of Zurich financed by an Ambizione Grant of the Swiss National Science Foundation (SNSF). Nevertheless, one main result and several partial results could be obtained.
