Mesoscopic Effects in Bose-Einstein Condensate Fluctuations of an Ideal Gas in a Box

The mesoscopic effects in the quantum trapped gases of the Bose atoms constitute the main subject of the present thesis. These effects are the most difficult for the theoretical analysis in the quantum statistical physics since they can?t be seen by neither a standard quantum mechanics of the simple microscopic systems of one or very few atoms nor a standard statistical physics of the macroscopic systems that are infinite in the bulk (thermodynamic) limit. Most of the experiments on the cold quantum gases performed in the last decade, starting from the first demonstration of BEC in 1995, involve the mesoscopic systems of a finite number of atoms. The mesoscopic effects should manifest themselves most clearly and easily near a critical temperature of BEC; however, they could be observed also above and below the critical temperature. Here I study the quantum and thermal fluctuations of the Bose-Einstein condensate (BEC) in a box with the periodic boundary conditions under a particle-number constraint. The above constraint is the only reason for the BEC and is crucial for the mesoscopic effects in the BEC fluctuations, especially in the vicinity of the critical temperature in the Bose gas. I employ the particle-number conserving operator formalism of Girardeau and Arnowitt introduced in 1959 to analyze the canonical ensemble fluctuations. I present analytical formulas and numerical calculations for the central moments of the ground state occupation fluctuations in an ideal Bose gas in a box with a mesoscopic number of particles. I present the analysis of the BEC statistics both on a temperature at a fixed number of particles and on a number of particles at a fixed temperature. Both analyses are valid for the purpose of understanding the important mesoscopic effects near the critical temperature. I emphasize the non-Gaussian nature of the fluctuations. The presented formalism can be generalized to the case of a weakly interacting Bose gas in a box in the framework of the Bogoliubov approximation. The work in this direction is in progress but is not included in the present thesis.