Modern problems in Statistical Physics of Bose-Einstein Condensation and in Electrodynamics of Free Electron Lasers

In this dissertation, I have studied theoretical problems in statistical physics and electrodynamics of Bose particles, namely, mesoscopic effects in statistics of Bose- Einstein condensate (BEC) of atoms and electromagnetic waveguide effects of planar Bragg structures in Free Electron Lasers. A mesoscopic system of a trapped gas of Bose atoms is the most difficult for the theoretical analysis in quantum statistical physics since it cannot be studied by neither a quantum mechanics of the simple microscopic systems of one or very few atoms nor a standard statistical physics of the macroscopic systems that implies a thermodynamic limit. I present analytical formulas and numerical calculations for the moments and cumulants of BEC fluctuations in both ideal and weakly interacting gas. I analyze the universal scaling and structure of the BEC statistics in a mesoscopic ideal gas in the critical region. I present an exactly solvable Gaussian model of BEC in a degenerate interacting gas and its solution that confirms the universality and constraint-cut-off origin of the strongly non-Gaussian BEC statistics. I consider a two-energy-level trap with arbitrary degeneracy of an upper level and find an analytical solution for the condensate statistics in a mesoscopic ideal gas. I show how to model BEC in real traps by BEC in the two-level or three-level traps. I study wave propagation in the open oversized planar Bragg waveguides, in particular, in a planar metal waveguide with corrugation. I show that a step perturbation in a corrugation phase provides a high selectivity over transverse modes. I present a new Free Electron Laser (FEL) amplifier scheme, in which the radiation is guided by the planar Bragg structure with slightly corrugated walls and a sheet electron beam is traveling at a significant angle to the waveguide axis. By means of nonlinear analysis, I demonstrate that the proposed scheme provides an effective mode filtration and control over the structure of the output radiation and allows one to achieve amplification up to 30 dB in the existing FEL machines.