The distribution of the moment of inertia for harmonically trapped noninteracting Bosons at finite temperature: Large deviations

We compute the full probability distribution of the moment of inertia I ∝ ∑ᵢ₌₁^N r⃗_i\,2 I ∝ ∑ i = 1 N r → i 2 of a gas of N N noninteracting bosons trapped in a harmonic potential V(r) = (1/2)\, m\, ω^2 r^2 V ( r ) = ( 1 / 2 ) m ω 2 r 2 , in all dimensions and at all temperature. The appropriate thermodynamic limit in a trapped Bose gas consists in taking the limit N→ ∞ N → ∞ and ω→ 0 ω → 0 with their product ρ = N ω^d ρ = N ω d fixed, where ρ ρ plays the role analogous to the density in a translationally invariant system. In this thermodynamic limit and in dimensions d;gt 1 d > 1 , the harmonically trapped Bose gas undergoes a Bose-Einstein condensation (BEC) transition as the density ρ ρ crosses a critical value ρ_c(β) ρ c ( β ) , where β β denotes the inverse temperature. We show that the nature of the condensation is different for 1 1 < d ≤ 2 and d;gt 2