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 temperatures. The appropriate thermodynamic limit in a trapped Bose gas consists in taking $N\to\infty$ and $\omega\to 0$ with their product $\rho = N\omega^d$ fixed, where $\rho$ plays a role analogous to 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 $\rho$ crosses a critical value $\rho_c(\beta)$, where $\beta$ denotes the inverse temperature. We show that the nature of the condensation is different for $1 2$. Near the condensation transition, for simplicity, we provide the analysis only for $d>2$. 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. In addition, on the condensed side, $\Phi(z)$ becomes independent of $\rho$ for $z 1$. This provides a real-space diagnostic for the BEC transition in the noninteracting Bose gas.