Energy fluctuation of a finite number of interacting bosons: A correlated many-body approach

Abstract

We calculate the energy fluctuation of a truly finite number of interacting bosons and study the role of interaction. Although the ideal Bose gas in thermodynamic limit is an exactly solvable problem and analytic expression of various fluctuation measures exists, the experimental Bose-Einstein condensation (BEC) is a nontrivial many-body problem. We employ a two-body correlated basis function and utilize the realistic van der Waals interaction. We calculate the energy fluctuation $(△{E}^{2})$ of the interacting trapped bosons and plot $\frac{△{E}^{2}}{{k}_{B}^{2}{T}^{2}}$ as a function of $\frac{T}{{T}_{c}}$. In the classical limit $△{E}^{2}$ is related to the specific heat per particle ${c}_{v}$ through the relation $△{E}^{2}={k}_{B}{T}^{2}{c}_{v}$. We have obtained a distinct hump in $\frac{△{E}^{2}}{{k}_{B}^{2}{T}^{2}}$ around the condensation point for three-dimesional harmonically trapped Bose gas when the particle number $N\ensuremath{\simeq}5000$ and above which corresponds to the second-order phase transition. However for finite-size interacting bosons ($N\ensuremath{\simeq}$ a few hundred) the hump is not sharp, and the maximum in $\frac{△{E}^{2}}{{k}_{B}^{2}{T}^{2}}$ can be interpreted as a smooth increase in the scaled fluctuation below ${T}_{c}$ and then a decrease above ${T}_{c}$. To illustrate the justification we also calculate ${c}_{v}$, which exhibits the same feature, which leads to the conjecture that for finite-sized interacting bosons phase transition is ruled out.