Boundary effects on Bose-Einstein condensation in ultra-static space-times

Abstract

The boundary effects on the Bose-Einstein condensation with a nonvanishing chemical potential on an ultra-static space-time are studied. High temperature regime, which is the relevant regime for the relativistic gas, is studied through the heat kernel expansion for both Dirichlet and Neumann boundary conditions. The high temperature expansion in the presence of a chemical potential is generated via the Mellin transform method as applied to the harmonic sums representing the free energy and the depletion coefficient. The effects of boundary conditions on the relation between the depletion coefficient and the temperature are analyzed. Both charged and neutral bosons are considered.

Non-relativistic limit of thermodynamics of Bose field in a static space-time and Bose–Einstein condensation

“…The validity of (92) for the more difficult case where h involves the Poisson kernel instead of the heat kernel was shown in the Appendix B of [11]. Our easier case of heat kernel dependent h can be treated by straightforwardly adapting the discussion given there for the Poisson kernel.…”

“…dπ a (t, x)dφ a (t, x). (11) Note that since Z is the statistical partition function there is no factor of i multiplying H in (10). Now we have…”

“…The nonrelativistic limit in question is a c → ∞ limit which is taken in a way that leaves the static background geometry intact. In the ultrarelativistic regime the analysis of the thermodynamics of a quantum field in a static background is well understood [1,3,4,6,7,8,9,10,11]. Here we aim to study the opposite limit.…”

“…5 we apply our results to the Bose-Einstein condensation in a static spacetime. We also discuss the ultrarelativistic case using the well known ultrarelativistic (high temperature) expansion of the free energy [1,3,4,6,7,8,10,11]. In the Appendix A we give the details of the computations described in Sec.…”

Virial coefficients expressed by heat kernel coefficients

“…For electromagnetic fields, the heat kernel expansion is an effective tool to study the Casimir effect for various geometries [26,27,28]. The thermodynamical properties of relativistic quantum gases and the Bose-Einstein condensation are also discussed in flat and curved space [19,29,30]. However, there are few works to systematically discuss the virial expansion in terms of the heat kernel expansion.…”