Partition function of the Bose-Einstein condensate dark matter and the modified Gross-Pitaevskii equation

Intending to describe the dark matter (DM) of dwarf galaxies, we concentrate on one model of the slowly rotating and gravitating Bose-Einstein condensate (BEC). For a deeper understanding of its properties, we calculate the partition function and compare the characteristics derived from it with the results based on the solution of Gross-Pitaevskii equation (GPE). In our approach, which uses Green's functions of spatial evolution operators, we formulate in a unified way the boundary conditions, important for applying the Thomas-Fermi approximation. Taking this into account, we revise some of the results obtained earlier. We also derive the spatial particle distribution, similar to the model with rotation, by using the deformation of commutation relations for a macroscopic wave function and modifying the GPE. It is shown that such an approach leads to the entropy inhomogeneity and makes the distribution dependent on temperature.