Math Courses

Condensation phenomena in classical and quantum statistical mechanics

by Giacomo Gradenigo (Gran Sasso Science Institute)

→ Europe/Rome
Description

Lecturer
Giacomo Gradenigo (GSSI)


Course description
The course is a follow up of the class "Mathematical Problems in Quantum Mechanics" held on February 2019. The key problem we are going to address is: what is the behaviour of a collection of many quantum particles? Which are the tools to describe many quantum degrees of freedom? 


Course content
The course will consist of three parts:

  1. We introduce the so-called "second quantization" of bosonic particles, starting from the many-body Schrodinger Equation and arriving to the concept of quantized field, which is the cornerstone of the whole theoretical physics of 19th Century. After having explained the "second quantization" of bosonic particles in the general case, i.e. the one where particles interact, we will specialize to the case of "non-interacting" bosonic particles and introduce the phenomenon of Bose-Einstein condensation, which takes place when many of them are squeezed into a small enough volume. For the Bose-Einstein condensation we will need to introduce quantum statistical mechanics, i.e. the partition function/sum (measure of the volume in phase space) of quantum particles states. Finally will be focused on the derivation of the Gross-Pitaevskii equation, which is the equation for a Bose-Einstein condensate in presence of a small amount of interactions between the quantum particles. 
     
  2. We will discuss the equivalence between statistical ensemble in the context of classical mechanics: microcanonical-canonical-grandcanonical. We will conclude by reconsidering the Bose-Einstein condensation as a phenomenon of non-equivalence between the grand-canonical and the canonical ensemble. 
     
  3. The third part of the class will be dedicated to the study of localization in the Discrete Non-Linear Schrodinger Equation, which is a lattice version of the Gross-Pitaevskii equation. We will point out the similarity between the mechanism of Bose-Einstein condensation and that of localization. 


References

  • "Quantum Theory of Many-Particle Systems" A. Fetter, J.-D. Walecka
  • "Statistical Mechanics", Kerson Huang,
  • "Mathematics of physicists", P. Dennery and A. Krzywicki