We study a system of 1D non-interacting spinless fermions in a
confining trap at finite temperature. We first derive a useful and
general relation for the fluctuations of the occupation numbers valid
for arbitrary confining trap, as well as for both canonical and grand
canonical ensembles. Using this relation, we obtain compact expressions,
in the case of the harmonic trap, for the variance of certain
observables of the form of sums of a function of the fermions’
positions, \mathcal{L}=\sum_n h(x_n)ℒ=∑nh(xn).
Such observables are also called linear statistics of the positions. As
anticipated, we demonstrate explicitly that these fluctuations do depend
on the ensemble in the thermodynamic limit, as opposed to averaged
quantities, which are ensemble independent. We have applied our general
formalism to compute the fluctuations of the number of fermions
\mathcal{N}_+𝒩+
on the positive axis at finite temperature. Our analytical results are
compared to numerical simulations. We discuss the universality of the
results with respect to the nature of the confinement.
Statistical properties of the momentum occupation numbers of the Tonks-Girardeau gas in a harmonic trap
