Bose-Einstein condensation of a Knudsen gas

We reconcile a long-standing controversy regarding the transition temperature of the Bose-Einstein condensation in a dilute interacting Bose gas, by showing that there is a crossover between ideal gas and interacting gas. The former corresponds to a Knudsen (or collisionless) regime, in which the mean-free-path is much larger than the system dimension, while the latter corresponds to the opposite hydrodynamic regime. The deviation of the transition temperature from that of the non-interacting gas is proportional to √ a in the Knudsen regime, and a in the hydrodynamic regime, where a is the scattering length. This crossover may be observable in a Bose gas trapped in a potential or on an optical lattice. The transition temperature Tc of a dilute interacting Bose gas in 3D has been a subject of much theoretical interest and controversy. Let us denote its fractional deviation from that of the non-interacting gas by ∆ ≡ Tc − T (0) c T (0) c , (1) where T (0) c = 2π~ mkB [nζ (3/2)] −2/3 , ζ (3/2) ≈ 2.612, n = N/L, with N the number of particles and L the linear size of the system. Using an argument based on the virial expansion in the high-temperature phase, the author [1] obtained the following result: