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

TL;DR

The paper derives the full finite-$T$ distribution of the moment of inertia $I$ for a harmonically trapped, ideal Bose gas in all dimensions, focusing on the thermodynamic limit with density $\rho=N\omega^d$ fixed. A duality relates the cumulant generating function of $I$ to two partition functions at related temperatures, enabling an exact large-$N$ analysis of the cumulant generating function $\Psi(\lambda)$ and the rate function $\Phi(z)$. In $d>1$ the BEC transition imprints a nonanalyticity in $\Phi(z)$ at a critical point $z_c$, with a discontinuous second derivative, while no such singularity exists for $d\le1$; this rate-function feature offers a real-space diagnostic of condensation even when the system remains in the fluid phase. The work provides explicit asymptotics for $\Phi(z)$ in multiple limits and highlights a crossover regime near the transition, demonstrating how rare fluctuations of $I$ encode the phase structure of the underlying quantum gas. These findings have potential implications for experimental probes of BEC through real-space measurements and contribute to the broader understanding of large deviations in quantum many-body systems.

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 $V(r) = (1/2)\, m\, ω^2 r^2$, in all dimensions and at all temperature. The appropriate thermodynamic limit in a trapped Bose gas consists in taking the limit $N\to \infty$ and $ω\to 0$ with their product $ρ= N ω^d$ fixed, where $ρ$ 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 $ρ$ crosses a critical value $ρ_c(β)$, where $β$ denotes the inverse temperature. We show that the probability distribution $P_β(I,N)$ of $I$ admits a large deviation form $P_β(I,N) \sim e^{-V Φ(I/V)}$ where $V = ω^{-d} \gg 1$. We compute explicitly the rate function $Φ(z)$ 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 $ρ< ρ_c(β)$, 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.

Figures (6)

• Figure 1: The solid blue line is a plot of the function $h_{d=3}(s) = {\rm Li}_{d=3}(e^{-s})$ given in Eq. (\ref{['sp_eq2']}) as a function of $s$. The value at $s=0$, namely $h_{d=3}(s=0) = \zeta(3)$, denotes the critical density $\tilde{\rho}_c$. The two horizontal lines indicates two representative values of the density $\tilde{\rho}$: one in the fluid phase ($\tilde{\rho} < \tilde{\rho}_c$) and the other in the condensed phase (where $\tilde{\rho} > \tilde{\rho}_c$). In the fluid phase, given the density $\tilde{\rho} < \tilde{\rho}_c$, there is a unique value $s^*$ such that $h_{d=3}(s^*) = \tilde{\rho}$ as shown in the figure. This shows that, in the fluid phase, there is a nonzero value $s=s^*$ where a saddle point occurs. As $\tilde{\rho}$ approaches $\tilde{\rho}_c$ from below, the saddle point $s^*$ approaches to $0$.
• Figure 2: The vertical blue line denotes the Bromwich contour $\Gamma$ in the complex $s$-plane in Eq. (\ref{['ZN_sp2']}). The Bromwich contour intersects the real axis at $s=s^*$ denoting the location of the saddle point in Eq. (\ref{['sp_eq2']}).
• Figure 3: Plot of the $\ln Z(N,\beta)/V$ (in the limit $V \to \infty$) vs $\rho$ close to the critical density $\rho = \rho_c(\beta)$ as given in Eq. (\ref{['summary']}). The curve decreases quadratically for $\rho<\rho_c(\beta)$ (shown by the blue line), and freezes to the constant $A_0 = \zeta(d+1)/\beta^d$ for $\rho>\rho_c(\beta)$ (as shown by the red horizontal line). In the inset, we zoom in in the region where $\rho - \rho_c(\beta) = O(1/\sqrt{V})$ and show schematically how the crossover occurs from the left (fluid) to the right (condensed), as obtained by taking the logarithm in Eq. (\ref{['Z_crossover']}).
• Figure 4: Phase diagram in the $(\lambda,\rho)$ plane. The allowed values of $\lambda$ lie in the range $\lambda \in (-\beta/2,+\infty)$ and $\rho \geq 0$. The black horizontal line represents the physical critical density $\rho = \rho_c(\beta)$, which is independent of $\lambda$. The solid blue curve represents $\rho_c(\tilde{\beta})$ with $\tilde{\beta} = \sqrt{\beta(\beta + 2 \lambda)}$, as given in Eq. (\ref{['rhoc_tilde']}) and it diverges as $\lambda \to -\beta/2$. These two curves meet at $(\lambda=0, \rho = \rho_c(\beta))$, marked by a red filled circle. The four colored phases representing the fluid-fluid (F-F), the fluid-condensed (F-C), the condensed-fluid (C-F) and the condensed-condensed (C-C) meet at the red filled circle.
• Figure 5: The function $\Psi(\lambda)$ plotted as a function of $\lambda$ for $\rho<\rho_c(\beta)$ (left panel) and $\rho>\rho_c(\beta)$ (right panel). In both panels, $\lambda_c$ marks the value of $\lambda$ that separates two phases as one increases $\lambda$ and the value of $\lambda_c$ is given in Eq. (\ref{['lmax']}). In the figure, we used $\beta = 1/2$. On the left panel, $\lambda_c$ separates the F-F and the F-C phases, while on the right panel it separates the C-F and the C-C phases. As $\lambda$ crosses $\lambda_c$, in both panels, the functions $\Psi(\lambda)$ and its first derivative are continuous, while the second derivative $\Psi"(\lambda)$ undergoes a discontinuous jump (though not visible in the figure).