Correlations of occupation numbers in the canonical ensemble and application to BEC in a 1D harmonic trap

TL;DR

This paper derives general formulas for occupation number correlations in the canonical ensemble for bosons and fermions, applying them to analyze Bose-Einstein condensation in a 1D harmonic trap, revealing quantum correlation effects.

Contribution

It introduces determinant and Schur function representations for correlation functions, providing analytical results for BEC in a 1D trap and occupation number distributions.

Findings

Correlation functions expressed as ratios of determinants

Analytical moments and correlations in 1D BEC

Ground state occupancy follows a truncated Gumbel distribution

Abstract

We study statistical properties of $N$ non-interacting identical bosons or fermions in the canonical ensemble. We derive several general representations for the $p$-point correlation function of occupation numbers $\overline{n_1\cdots n_p}$. We demonstrate that it can be expressed as a ratio of two $p\times p$ determinants involving the (canonical) mean occupations $\overline{n_1}$, ..., $\overline{n_p}$, which can themselves be conveniently expressed in terms of the $k$-body partition functions (with $k\leq N$). 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 1D 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.