Phase structures of the black D$p$-D$(p + 4)$-brane system in various ensembles II: electrical and thermodynamic stability

By incorporating the electrical stability condition into the discussion, we continue the study on the thermodynamic phase structures of the D$p$-D$(p + 4)$ black brane in GG, GC, CG, CC ensembles defined in our previous paper arXiv:1502.00261. We find that including the electrical stability conditions in addition to the thermal stability conditions does not modify the phase structure of the GG ensemble but puts more constraints on the parameter space where black branes can stably exist in GC, CG, CC ensembles. In particular, the van der Waals-like phase structure which was supposed to be present in these ensembles when only thermal stability condition is considered would no longer be visible, since the phase of the small black brane is unstable under electrical fluctuations. However, the symmetry of the phase structure by interchanging the two kinds of brane charges and potentials is still preserved, which is argued to be the result of T-duality.