S5B3 - Graduate Seminar on Partial Differential Equations in the Sciences - Energy and Dynamics of Boson Systems

Organizational Meeting : Monday, February 21. Details will follow.

Short Description of the Seminar : In the last years there was a lot of progress in the theoretical study of bosonic systems. In this seminar, we are going to review some of the mathematically rigorous results obtained in this area. Possible topics are as follows.

Energy of dilute Bose gases: in 1940, a celebrated paper by Bogoliubov proposed a formula for the ground state energy of a dilute Bose gas. Bogoliubov's formula is based on a physically reasonable assumption, which, however, is very difficult to motivate in mathematical rigorous terms. The proof of this formula was a challenge to mathematical physicists for over 50 years, until the problem was recently solved.

Energy of trapped Bose gases: experiments on Bose gases are typically conducted inside strong magnetic traps, at extremely low temperatures. In this regime, the system essentially relaxes to it ground state. For this reason, it is important to understand the ground state properties and, in particular, the ground state energy of trapped Bose gases. It turns out that the many-body ground state energy is given, in first approximation, by minimization of appropriate one-particle energy functionals.

Bose-Einstein condensation for trapped Bose gases: Bose-Einstein condensates are states of a many boson system where a macroscopic number of particles (a number of particles proportional to the total number) is described by the same one-particle wave function. Bose and Einstein proved, already in the Twenties, the existence of condensation at sufficiently low temperature for non-interacting boson systems. To prove the existence of Bose-Einstein condensation for interacting systems is, in general, a major open problem in mathematical physics. There is, however, a particular class of interacting systems, characterized by an extremely low density, for which the existence of Bose-Einstein condensation can be established rigorously.

Quantum dynamics in the mean field limit: typical systems of interest in physics are characterized by a huge number N of interacting particles. For such huge values of N, it is essentially impossible to solve the time-dependent Schroedinger equation, neither analytically nor numerically. For this reason, one of the main questions of non-equilibrium statistical mechanics is the derivation of effective evolution equations which approximate, in the relevant regimes, the solution to the many-body Schroedinger equation. It turns out that the derivation of effective evolution equation is particularly simple in the so called mean field regime, where particles experience many, very weak, collisions. In this regime, the many body evolution of a condensate can be approximated, in an appropriate sense, by a nonlinear one-particle Hartree equation.

Derivation of the Gross-Pitaevskii equation: when very dilute trapped gases are released (when traps are switched off), they begin to evolve. It turns out that the time-evolution of the initially trapped condensates can be described by the nonlinear Gross-Pitaevskii equation. The understanding of the dynamics in this limit is much more difficult compared with the mean-field limit, because here interactions are very rare (the gas is dilute) but, at the same time, very strong. For this reason, correlations among particles play here a crucial role.

Some relevant literature :

Lieb, Seiringer, Solovej, Yngvason. The mathematics of the Bose gas and its condensation. Oberwolfach Seminars, Band 34.

Schlein. Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics. Lecture Notes for 2008 Clay Summer School. Preprint arXiv:0807.4307.