Abstract
The first lecture was devoted to a simple problem, the study of the equilibrium state of a perfect gas composed of indistinguishable particles. The starting point was the work of Bose, who first tackled this problem in his studies of blackbody radiation, i.e. a thermal gas of photons. Einstein's generalization to a gas of material particles was almost immediate, and introduced the concept of saturation of excited states: for a system of finite size and given temperature, the number of particles that can be placed in the set of energy levels outside the fundamental level is bounded. Particles in excess of this limit can only occupy the fundamental level, forming a Bose-Einstein condensate.
Once this saturation has been achieved, for a gas confined in a box or in a harmonic trap, we have studied how to take the thermodynamic limit of the system, by tending the size of the gas towards infinity while keeping the intensive parameters (density, temperature) constant. We have shown that the "survival" of saturation depends on both the dimensionality of the space and the nature of the gas confinement. We illustrated these different concepts with recent experimental results, obtained in particular with "flat-bottom" traps. We concluded this lecture with a first-ever discussion of Bose-Einstein condensation beyond the perfect gas, using the criterion proposed by Penrose and Onsager.
