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

Abstract

We compute the full probability distribution of the moment of inertia $I \propto \sum_{i=1}^N \vec{r}_i^{\,2}$ of a gas of $N$ noninteracting bosons trapped in a harmonic potential $V(r) = (1/2)\, m\, \omega^2 r^2$, in all dimensions and at all temperature. The appropriate thermodynamic limit in a trapped Bose gas consists in taking the limit $N\to \infty$ and $\omega\to 0$ with their product $\rho = N \omega^d$ fixed, where $\rho$ plays the role analogous to the density in a translationally invariant system. In this thermodynamic limit and in dimensions $d>1$, the harmonically trapped Bose gas undergoes a Bose-Einstein condensation (BEC) transition as the density $\rho$ crosses a critical value $\rho_c(\beta)$, where $\beta$ denotes the inverse temperature. We show that the probability distribution $P_\beta(I,N)$ of $I$ admits a large deviation form $P_\beta(I,N) \sim e^{-V \Phi(I/V)}$ where $V = \omega^{-d} \gg 1$. We compute explicitly the rate function $\Phi(z)$ and show that it exhibits a singularity at a critical value $z=z_c$ where its second derivative undergoes a discontinuous jump. We show that the existence of such a singularity in the rate function is directly related to the existence of a BEC transition and it disappears when the system does not have a BEC transition as in $d \leq 1$. An interesting consequence of our results is that even if the actual system is in the fluid phase, i.e., when $\rho < \rho_c(\beta)$, by measuring the distribution of $I$ and analysing the singularity in the associated rate function, one can get a signal of the BEC transition in $d>1$. This provides a real space diagnostic for the BEC transition in the noninteracting Bose gas.