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

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.