Abstract
When the D$(p+4)$-brane ($p=0,1,2$) with delocalized D$p$ charges is put into equilibrium with a spherical thermal cavity, the two kinds of charges can be put into canonical or grand canonical ensemble independently by setting different conditions at the boundary. Using the thermal stability condition, we discuss the phase structures of various ensembles of this system formed in this way and find out the situations that the black brane could be the final stable phase in these ensembles. In particular, van der Waals-like phase transitions can happen when D0 and D4 charges are in different kinds of ensembles. Furthermore, our results indicate that the D$(p+4)$-branes and the delocalized D$p$-branes are equipotent.
Highlights
As the solutions of the supergravity, black branes, like black holes, can have their own thermodynamics
Our results indicate that the D(p + 4)-branes and the delocalized Dp-branes are equipotent
Study on the thermodynamical phase structure of the black branes is valuable in understanding the non-perturbative nature of String theory
Summary
We consider a gravitational system bounded by a big spherical reservoir which can be regarded as a spherical boundary at the transverse radius ρb. The third and fourth terms come from the Dp - and D(p + 4)-brane form field action, Ip. Iboundary could admit several boundary integration terms, which depends on what ensemble we are interested in. Γ in these equations is the determinant of the induced metric on the (4 − p)-dimensional boundary, and the (p + 1)-form and (p + 5)-form fields A[p+1] and A[p+5] are the RamondRamond potentials of Dp - and D(p + 4)-branes respectively. Boundary terms when doing variations with respect to the gauge field potentials. If we fix the gauge field strength F on the boundary and the potential A are not fixed, after partial integration the additional boundary terms would emerge, we need to include Ib,p or Ib,p+4 whose variations would cancel these terms. The relation between boundary conditions and ensembles will be addressed in subsection
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