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 $I\propto {\sum }_{i=1}^N{\vec{r}}_i^2$ of a gas of $N$ noninteracting bosons trapped in a harmonic potential $V(r)=(1\mathrm{/}2)m{\omega }^2{r}^2$, in all dimensions and at all temperature. The appropriate thermodynamic limit in a trapped Bose gas consists in taking the limit $N\to \mathrm{\infty }$ 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 $P_{\beta }(I,N)\sim e^{-V\mathrm{\Phi }(I\mathrm{/}V)}$ where $V={\omega }^{-d}\gg 1$. We compute explicitly the rate function $\mathrm{\Phi }(z)$ 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 $d\le 1$. 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.