The intrinsic curvature of thermodynamic potentials for black holes with critical points

The geometry of thermodynamic state space is studied for asymptotically anti-de Sitter black holes in D-dimensional space times. Convexity of thermodynamic potentials and the analytic structure of the response functions is analysed. The thermodynamic potentials can be used to define a metric on the space of thermodynamic variables and two commonly used such metrics are the Weinhold metric, derived from the internal energy, and the Ruppeiner metric, derived from the entropy. The intrinsic curvature of these metrics is calculated for charged and for rotating black holes and it is shown that the curvature diverges when heat capacities diverge but, contrary to general expectations, the singularities in the Ricci scalars do not reflect the critical behaviour. When a cosmological constant is included as a state space variable it can be interpreted as a pressure and the thermodynamically conjugate variable as a thermodynamic volume. The geometry of the resulting extended thermodynamic state space is also studied, in the context of rotating black holes, and there are curvature singularities when the heat capacity at constant angular velocity diverges and when the black hole is incompressible. Again the critical behaviour is not visible in the singularities of the thermodynamic Ricci scalar.Comment: 35 pages, 3 figure

Bah Humbug: Unexpected Christmas Cards and the Reciprocity Norm

The reciprocity norm refers to the expectation that people will help those who helped them. A well-known study revealed that the norm is strong with Christmas cards, with 20% of people reciprocating a Christmas card received from a stranger. I attempted to conceptually replicate and extend this effect. In Study 1, 755 participants received a Christmas card supposedly from a more- versus less-similar stranger. The reciprocation rate was unexpectedly low (2%), which did not allow for a test of a similarity effect. Two potential reasons for this low rate were examined in Study 2 in which 494 participants reported their likelihood of reciprocating a Christmas card from a stranger as well as their felt suspicions/threat about the card and their frequency of e-mail use. Reciprocation likelihood was negatively correlated with perceived threat/suspicion and e-mail use. It appears that reciprocating a gift from a stranger in offline settings may be less likely than expected

Bose condensation and branes

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 $AdS_5\times S^5$, for ${\cal N}=4$ 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.Comment: 9 pages, 1 figur

Potential Flow Of The Renormalisation Group In A Simple Two Component Model

The renormalisation group (RG) flow on the space of couplings of a simple model with two couplings is examined. The model considered is that of a single component scalar field with $\phi^4$ self interaction coupled, via Yukawa coupling, to a fermion in flat four dimensional space. The RG flow on the two dimensional space of couplings, ${\cal G}$, is shown to be derivable from a potential to sixth order in the couplings, which requires two loop calculations of the $\beta$-functions. The identification of a potential requires the introduction of a metric on ${\cal G}$ and it is shown that the metric defined by Zamalodchikov, in terms of two point correlation functions of composite operators, gives potential flow to this order.Comment: 7 pages Typset in PlainTeX, C Version 3.14