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

TL;DR

This paper derives the full probability distribution of the moment of inertia for a harmonically trapped noninteracting Bose gas at finite temperature, revealing a singularity in the large deviation rate function that signals Bose-Einstein condensation.

Contribution

It explicitly computes the large deviation rate function for the moment of inertia in all dimensions and connects its singularity to the BEC transition, providing a new real-space diagnostic.

Findings

The distribution follows a large deviation form with an explicit rate function.

A singularity in the rate function indicates the BEC transition in dimensions greater than one.

The singularity disappears in one or fewer dimensions where no BEC occurs.

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.