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Within an exact microcanonical (MC) ensemble, we study the nonanalyticities of thermodynamic functions research in finite noninteracting Bose gases in traps. The results show that there exists a rich oscillatory behavior of MC thermodynamical quantities as a function of a system's total energy E (e.g., nonmonotonous temperature, nonanalytic and negative specific heats, and microscopic phase transitions). The origin of these nonanalyticities comes directly from the inverted curvature entropy S(E) with respect to E and the behaviors are different in different trap geometries, boundary conditions, and energy spectrum configurations. Contrary to the usual grandcanonical and canonical results, there exists Bose condensation and the nonanalyticities in the two-dimensional finite noninteracting Bose systems with different traps. We also discuss the critical temperature dependence on the particle number N with different ensembles, traps, and boundary conditions. In large enough N, almost all the results of the thermodynamical quantities become smooth, which are similar to the usual canonical behaviors. We emphasize the finite-size effects on the MC entropy change, which should, in principle, be observable in suitably designed experiments of the small systems.

Based on a new statistical theory, we investigate the thermodynamic anomalies of small quantum systems, such as the negative specific heat (NSH) and negative entropy (NE) within the generalized canonical ensemble. We consider the systembath heat exchange and "uncompensated heat" in the thermodynamical level which is independent on the details of the system-bath coupling. For ideal finite systems, we calculate two thermodynamic quantities, i.e., the experimental specific heat and the entropy. The results show that the NSH and NE exist in quantum thermodynamics, particularly at low temperatures for small systems. They agree with the results of the reduced partition function theory and reveal that the finite boundary effects of the uncompensated heat and heat exchange of small quantum systems dominate the nonequilibrium irreversible processes. Keywords Small quantum system · Negative specific heat · Negative entropy Thermodynamics and the traditional statistical-mechanical interpretation are originally developed for large systems in the equilibrium state. However, the theories are not sufficient to describe the thermodynamical properties of small quantum systems [1-3]. There are increasing findings of anomalies presented experimentally in these systems [4][5][6][7][8][9]. These experimental results [6,10,11] turn up the existence of negative specific heat (NSH) and negative entropy (NE) [12]. But the specific heat is necessarily positive both in the standard Boltzmann-Gibbs statistical theories of the canonical ensemble (CE) and grand-canonical ensemble (GCE) and in the exact CE theory on the