Thermodynamic quantities, occupation numbers and their fluctuations of a one-dimensional Bose gas confined by a harmonic potential are studied using different ensemble approaches. Combining number theory methods, a new approach is presented to calculate the occupation numbers of different energy levels in microcanonical ensemble. The visible difference of the ground state occupation number in grand-canonical ensemble and microcanonical ensemble is found to decrease by power law as the number of particles increases.

Contrary to the common thermodynamic systems, systems with long-range interaction may cause nonconcavity of s(ε) curves. In this paper, we propose a long-range interacting Ising chain in a staggered magnetic field model which has a non-concave entropy part. In this model, the first phase transition is accompanied by the phenomenon of temperature jump in microcanonical ensemble when proper magnetic field intensity is met, while this jump cannot be observed in canonical ensemble, which shows the non-equivalence of different ensembles.To exhibit the cross-over process from microcanonical to canonical, the cross-over phase transition properties areexhibited recently by putting the chain in thermal contact with an adjustable two-level heat reservoir. In this paper, we introduce a different method by employing Gaussian ensemble to show the cross-over process reversely, i.e.,from canonical to microcanonical ensembles. As shown in this paper, by adjusting the parameters of the supporting parabolas in the Gaussian ensemble, one can observe the caloric curve of the system in any Gaussian ensemble.