The distribution of the moment of inertia for harmonically trapped noninteracting Bosons at finite temperature: Large deviations
4 affiliations
- 1Tata Institute of Fundamental Research
- 2Laboratoire de Physique Théorique et Modèles Statistiques
- 3University of Paris-Saclay
- 4Laboratory of Theoretical and High Energy Physics
Abstract
We compute the full probability distribution of the moment of inertia I \propto \sum_{i=1}^N \vec{r}_i^{\,2} I ∝ ∑ i = 1 N r → i 2 of a gas of N N noninteracting bosons trapped in a harmonic potential V(r) = (1/2)\, m\, \omega^2 r^2 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 ∞ N → ∞ and \omega\to 0 ω → 0 with their product \rho = N \omega^d ρ = N ω d fixed, where \rho ρ plays the role analogous to the density in a translationally invariant system. In this thermodynamic limit and in dimensions d;gt 1 d > 1 , the harmonically trapped Bose gas undergoes a Bose-Einstein condensation (BEC) transition as the density \rho ρ crosses a critical value \rho_c(\beta) ρ c ( β ) , where \beta β denotes the inverse temperature. We show that the nature of the condensation is different for 1<d≤ 2 1 < d ≤ 2 and d;gt 2 d > 2 . Near the condensation transition, for simplicity, we provide the analysis only for d;gt 2 d > 2 . We show that the probability distribution P_\beta(I,N) P β ( I , N ) of I I admits a large deviation form P_\beta(I,N) \sim e^{-V \Phi(I/V)} P β ( I , N ) ∼ e − V Φ ( I / V ) where V = \omega^{-d} \gg 1 V = ω − d ≫ 1 . We compute explicitly the rate function \Phi(z) Φ ( z ) and show that it exhibits a singularity at a critical value z=z_c z = z c where its second derivative undergoes a discontinuous jump. In addition, on the condensed side, the rate function \Phi(z) Φ ( z ) becomes independent of \rho ρ for z<z_c z < z c . We show that the existence of z_c z c and the associated freezing below z_c z c 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 ≤ 1 d ≤ 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) ρ < ρ c ( β ) , by measuring the distribution of I I and analysing the singularity in the associated rate function, one can get a signal of the BEC transition in d;gt 1 d > 1 . This provides a real space diagnostic for the BEC transition in the noninteracting Bose gas.
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BibTeX
@article{kulkarni2026distribution,
title = {{The distribution of the moment of inertia for harmonically trapped noninteracting Bosons at finite temperature: Large deviations}},
author = {Manas Kulkarni and Satya N. Majumdar and Grégory Schehr},
journal = {SciPost},
year = {2026},
month = oct,
doi = {10.21468/scipostphys.21.4.102},
url = {http://scipost.org/SciPostPhys.21.4.102},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - The distribution of the moment of inertia for harmonically trapped noninteracting Bosons at finite temperature: Large deviations
AU - Manas Kulkarni
AU - Satya N. Majumdar
AU - Grégory Schehr
PY - 2026
DA - 2026/10/22
JO - SciPost
DO - 10.21468/scipostphys.21.4.102
UR - http://scipost.org/SciPostPhys.21.4.102
AB - We compute the full probability distribution of the moment of inertia I \propto \sum_{i=1}^N \vec{r}_i^{\,2} I ∝ ∑ i = 1 N r → i 2 of a gas of N N noninteracting bosons trapped in a harmonic potential V(r) = (1/2)\, m\, \omega^2 r^2 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 ∞ N → ∞ and \omega\to 0 ω → 0 with their product \rho = N \omega^d ρ = N ω d fixed, where \rho ρ plays the role analogous to the density in a translationally invariant system. In this thermodynamic limit and in dimensions d;gt 1 d > 1 , the harmonically trapped Bose gas undergoes a Bose-Einstein condensation (BEC) transition as the density \rho ρ crosses a critical value \rho_c(\beta) ρ c ( β ) , where \beta β denotes the inverse temperature. We show that the nature of the condensation is different for 1<d≤ 2 1 < d ≤ 2 and d;gt 2 d > 2 . Near the condensation transition, for simplicity, we provide the analysis only for d;gt 2 d > 2 . We show that the probability distribution P_\beta(I,N) P β ( I , N ) of I I admits a large deviation form P_\beta(I,N) \sim e^{-V \Phi(I/V)} P β ( I , N ) ∼ e − V Φ ( I / V ) where V = \omega^{-d} \gg 1 V = ω − d ≫ 1 . We compute explicitly the rate function \Phi(z) Φ ( z ) and show that it exhibits a singularity at a critical value z=z_c z = z c where its second derivative undergoes a discontinuous jump. In addition, on the condensed side, the rate function \Phi(z) Φ ( z ) becomes independent of \rho ρ for z<z_c z < z c . We show that the existence of z_c z c and the associated freezing below z_c z c 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 ≤ 1 d ≤ 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) ρ < ρ c ( β ) , by measuring the distribution of I I and analysing the singularity in the associated rate function, one can get a signal of the BEC transition in d;gt 1 d > 1 . This provides a real space diagnostic for the BEC transition in the noninteracting Bose gas.
SN - 2542-4653
ER -