Bose condensation and branes

Abstract

When the cosmological constant is considered to be a thermodynamical variable in black hole thermodynamics, analogous to a pressure, its conjugate variable can be thought of as a thermodynamic volume for the black hole. In the AdS/CFT correspondence this interpretation cannot be applied to the CFT on the boundary but, from the point of view of the boundary SU(N ) gauge theory, varying the cosmological constant in the bulk is equivalent to varying the number of colors in the gauge theory. This interpretation is examined in the case of AdS 5 × S 5 , for N = 4 SUSY Yang-Mills at large N , and the variable thermodynamically conjugate to N , a chemical potential for color, is determined. It is shown that the chemical potential in the high temperature phase of the Yang-Mills theory is negative and decreases as temperature increases, as expected. For spherical black holes in the bulk the chemical potential approaches zero as the temperature is lowered below the Hawking-Page temperature and changes sign at a temperature that is within one part in a thousand of the temperature at which the heat capacity diverges.

Phase transition and thermodynamical geometry for Schwarzschild AdS black hole in AdS5 × S5 spacetime

“…Very recently, it was argued that it is more suitable to view the cosmological constant as the number of colors in gauge field and its conjugate as associated chemical potential [45][46][47]. This interpretation was examined in the case of AdS 5 × S 5 , for N = 4 supersymmetric Yang-Mills theory at large N in [45]. The chemical potential conjugate to the number of colors, is calculated.…”

“…In next section, we will review some basic thermodynamic properties of a black hole in AdS 5 × S 5 spacetime by treating the cosmological constant in the bulk as the number of colors [45]. In section 3 we will calculate the thermodynamical curvatures of the Weinhold metric, Ruppeiner metric and Quevedo metric, respectively, for the thermodynamical system, in order to see the relations between the thermodynamical curvature and phase transition.…”

“…[45]. In AdS 5 × S 5 spacetime, the line element for a five-dimensional Schwarzschild AdS black hole reads [10] …”

“…This is quite different from the classical gas with negative chemical potential. When the chemical potential approaches zero and becomes positive, quantum effects should come into playing some role [45]. Now we turn to the thermodynamical geometry of the black hole and want to see whether the thermodynamical curvature can reveal the singularity of these two heats capacities.…”

Black holes and Boyle's law — The thermodynamics of the cosmological constant

“…The thermodynamic variable conjugate to Λ would then act as a chemical potential, µ, for color. This chemical potential was calculated, for N = 4 SUSY Yang-Mills at large N , in [140] and shown to have the general properties that a chemical would be expected to have, for example at large T it is negative and is a decreasing function of T . For flat event horizons µ remains negative at small T while for spherical event horizons it can pass through zero and become positive at a temperature that is only some 6% below the de-confining phase transition of the gauge theory proposed in [139].…”