Partition Function of the Bose-Einstein Condensed Dark Matter and the Modified Gross-Pitaevskii Equation

TL;DR

This paper explores the properties of a gravitating Bose-Einstein condensate as a dark matter model for dwarf galaxies by calculating its partition function and modifying the Gross-Pitaevskii equation to include temperature effects.

Contribution

It introduces a unified approach to boundary conditions using Green's functions and revises previous results, incorporating temperature-dependent entropy and spatial distribution modifications.

Findings

Partition function calculation for the condensate

Modified Gross-Pitaevskii equation including temperature effects

Revised boundary conditions using Green's functions

Abstract

Intending to describe the dark matter of dwarf galaxies, we concentrate on one model of the slowly rotating and gravitating Bose-Einstein condensate. 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. In our approach, which uses the 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 Gross-Pitaevskii equation. It is shown that such an approach leads to the entropy inhomogeneity and makes the distribution dependent on temperature.