Phase structure of black branes in grand canonical ensemble

Abstract

This is a companion paper of our previous work [1] where we studied the thermodynamics and phase structure of asymptotically flat black $p$-branes in a cavity in arbitrary dimensions $D$ in a canonical ensemble. In this work we study the thermodynamics and phase structure of the same in a grand canonical ensemble. Since the boundary data in two cases are different (for the grand canonical ensemble boundary potential is fixed instead of the charge as in canonical ensemble) the stability analysis and the phase structure in the two cases are quite different. In particular, we find that there exists an analog of one-variable analysis as in canonical ensemble, which gives the same stability condition as the rather complicated known (but generalized from black holes to the present case) two-variable analysis. When certain condition for the fixed potential is satisfied, the phase structure of charged black $p$-branes is in some sense similar to that of the zero charge black $p$-branes in canonical ensemble up to a certain temperature. The new feature in the present case is that above this temperature, unlike the zero-charge case, the stable brane phase no longer exists and `hot flat space' is the stable phase here. In the grand canonical ensemble there is an analog of Hawking-Page transition, even for the charged black $p$-brane, as opposed to the canonical ensemble. Our study applies to non-dilatonic as well as dilatonic black $p$-branes in $D$ space-time dimensions.Comment: 32 pages, 2 figures, various points refined, discussion expanded, references updated, typos corrected, published in JHEP 1105:091,201

Modulating the phase structure of black D6 branes in canonical ensemble

“…We have recently found [10,11] that the same characteristic phase structure (including a Hawking-Page transition) of stabilized uncharged (Schwarzschild) black holes was also shared by the stabilized uncharged black p-branes in D-dimensional asymptotically flat spacetime (with the brane worldvolume dimensions d = 1 + p ) in string/M theory. However, in the charged case, the van der Waals-Maxwell liquid-gas type phase structure mentioned earlier, with its characteristic behavior 4 shown in Fig.…”

Phase structures of the black Dp-D(p + 4)-brane system in various ensembles I: thermal stability

“…the (35) in [35], except that now the solutionx depends on a new parameterΦ. From (3.16), we see that there is a turning point…”

“…Note that when Q = 0, the whole system becomes a D(p+4)-brane system, and the various phase structures of this system have already been thoroughly analyzed in [5,35]. Thus, in the following calculations, we will always assume that Q > 0 (Φ > 0).…”

“…In GG ensemble, we need all the three equations. For simplicity, in this paper we constraint ourselves only to the stability of branes under the change of the horizon size as in [6,13,35]. As will be shown later, this is equivalent to the thermal stability condition that the specific heat capacity is positive for the equilibrium state.…”

“…We will also give a brief discussion in a later section on the other electrical stability conditions like those used in [7] and leave the details in a paper to appear [31]. So here we will assume that the second and third equations in (3.1) have already been satisfied [13,35], and Q and q can be solved and be substituted in the first equation. Thus, the only problem left is to solve the first equation where b is a function of just one variable x, that is,…”