In this note an ideal Bose-Einstein and an ideal Fermi-Dirac gas are considered in the "limit
L," i.e., in the limit in which the number of particles
N and the volume
V of the system are allowed to tend to infinity so as to keep
VN a finite and nonzero constant. It is shown that, whatever the system of energy levels, the mean occupation numbers calculated on the basis of the grand canonical ensemble are in the limit
L the same as those calculated on the basis of the canonical ensemble. The only requirement is that the mean total number of particles for the grand canonical ensemble be equal to the fixed total number of particles for the canonical ensemble.
We study statistical properties of
N noninteracting identical bosons or fermions in the canonical ensemble. We derive several general representations for the
p-point correlation function of occupation numbers $\overline{{n}_{1}\ensuremath{\cdots}{n}_{p}}$. We demonstrate that it can be expressed as a ratio of two $p\ifmmode\times\else\texttimes\fi{}p$ determinants involving the (canonical) mean occupations
n1, ...,
np, which can themselves be conveniently expressed in terms of the
k-body partition functions (with $k\ensuremath{\le}N$). We draw some connection with the theory of symmetric functions and obtain an expression of the correlation function in terms of Schur functions. Our findings are illustrated by revisiting the problem of Bose-Einstein condensation in a one-dimensional harmonic trap, for which we get analytical results. We get the moments of the occupation numbers and the correlation between ground-state and excited-state occupancies. In the temperature regime dominated by quantum correlations, the distribution of the ground-state occupancy is shown to be a truncated Gumbel law. The Gumbel law, describing extreme-value statistics, is obtained when the temperature is much smaller than the Bose-Einstein temperature.