Microcanonical fluctuations of the condensate in weakly interacting Bose gases

We study fluctuations of the number of Bose condensed atoms in a weakly interacting homogeneous and trapped gases. For a homogeneous system we apply the particle-number-conserving formulation of the Bogoliubov theory and calculate the condensate fluctuations within the canonical and the microcanonical ensembles. We demonstrate that, at least in the low-temperature regime, predictions of the particle-number-conserving and traditional, nonconserving theory are identical, and lead to the anomalous scaling of fluctuations. Furthermore, the microcanonical fluctuations differ from the canonical ones by a quantity which scales normally in the number of particles, thus predictions of both ensembles are equivalent in the thermodynamic limit. We observe a similar behavior for a weakly interacting gas in a harmonic trap. This is in contrast to the trapped, ideal gas, where microcanonical and canonical fluctuations are different in the thermodynamic limit.

Figure 1

Fluctuations of the number of excited particles in the microcanonical and canonical ensemble for the system of $N=1000$ noninteracting bosons confined in a box with periodic boundary conditions (upper panel) and in a harmonic trap (lower panel). Analytical results (solid line) derived in MDE approximation are compared to the exact numerical data (dashed line). The temperature is scaled in the units of the critical temperature $T_C$.

Figure 2

Fluctuations of the number of noncondensed atoms in the canonical ensemble for interacting Bose gas confined in a box with periodic boundary conditions. Plots are shown for $\alpha =2.4$ and $\alpha =10$ (inset), where $\alpha =2Na∕\pi L$. The analytical result of Eq. (36) is compared with its low and high temperature expansions, Eqs. (39, 38), respectively. The temperature is scaled in the units of $\Delta =2{\pi }^2{\hslash }^2∕m{L}^2$.

Figure 3

Fluctuations of the number of noncondensed atoms in the canonical and microcanonical ensemble for a three-dimensional box with periodic boundary conditions. Plots are shown for $\alpha =2.4$ and $\alpha =10$ (inset), where $\alpha =2Na∕\pi L$. Depicted are the analytical result of Eqs. (36, 53) for the canonical and microcanonical fluctuations, respectively, the canonical fluctuations calculated numerically from Eq. (31), and the microcanonical fluctuations evaluated numerically with the help of recurrence algorithms. The temperature is scaled in the units of $\Delta =2{\pi }^2{\hslash }^2∕m{L}^2$.