In this thesis we study mathematical aspects of BCS theory and its connection to Ginzburg-Landau theory. Furthermore we investigate the exponential decay of solutions to the Kac master equation. A full abstract is contained in the document.
The thesis analyze Fermionic neutral systems with tunable attractive interaction between the particles whose strenght can be tuned from BCS theory to Bose-Einstein condensate. Below the critical temperature both systems are characterized by an order parameter. A mean-field approach can't describe correctly the phenomenon. This happens if Gaussian Thermal fluctations are included. The aim is to study BCS-BEC crossover around critical temperature through Ginzburg-Landau theory in 3D systems and Josephson effect in 2D systems. In both cases effect of thermal fluctuations have been included.