Thermodynamic properties of ideal Bose gas trapped in an external generic power law potential are investigated systematically from the grand thermodynamic potential in d dimensional space. The most general conditions for BoseEinstein condensate and the discontinuous conditions of heat capacity at the critical temperature in presence of generic power law potential are presented in this manuscript. The dependence of the physical quantities on external potential, particle characteristics and space dimensionality are discussed. The more general results obtained in this paper presents a unified illustration of Bose-Einstein condensation of ideal Bose systems as they reduces to the expressions and conclusions available in the literature with appropiate choice of power law exponent.

A unified description for the Bose and Fermi gases trapped in an external generic power law potential U = d i=1 c i | x i a i | n i is presented using the grandpotential of the system in d dimensional space. The thermodynamic quantities of the quantum gases are derived from the grand potential. An equivalence between the trapped Bose and Fermi gases is constructed from the thermodynamic quantities in one dimension (d = 1) using the Landen relation. It is also found that the established equivalence between the ideal free Bose and Fermi gases in d = 2 (Lee in Phys Rev E 55:1518, 1997) is lost when external potential is applied.

We study the Casimir eect for a three dimensional system of ideal free massive Bose gas in a slab geometry with Zaremba and anti-periodic boundary conditions. It is found that for these type of boundary conditions the resulting Casimir force is repulsive in nature, in contrast with usual periodic, Dirichlet or Neumann boundary condition where the Casimir force is attractive (Martin and Zagrebnov 2006 Europhys. Lett. 73 15). Casimir forces in these boundary conditions also maintain a power law decay function below condensation temperature and exponential decay function above the condensation temperature albeit with a positive sign, identifying the repulsive nature of the force.

We argue that, in the presence of time-dependent fluxes and quantum corrections, four-dimensional de Sitter solutions should appear in the type IIB string landscape and not in the swampland. Our construction considers generic choices of local and non-local quantum terms and satisfies the no-go and the swampland criteria, the latter being recently upgraded using the trans-Planckian cosmic censorship. Interestingly, both time-independent Newton constant and moduli stabilization may be achieved in such backgrounds even in the presence of time-dependent fluxes and internal spaces. However, once the time-dependence is switched off, any four-dimensional solution with de Sitter isometries appears to have no simple effective field theory descriptions and is back in the swampland.

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