On the applicability of holography in thermodynamic equilibrium

For a strongly coupled system that has a gravity dual description, we show that the standard holographic dictionary yields a nonnegative susceptibility when the system is in thermodynamic equilibrium and the correlation function is absolutely integrable. When the system has no spontaneous condensation or has a spontaneous ${\mathbb{Z}}_2$-symmetry breaking, we find that the trace energy condition" is violated in many cases. There is a normalized grand potential density that is monotonic as accessing to lower scales, providing a candidate $c$-function characterizing the number of effective degrees of freedom. Finally, we discuss a paradox’’ raising by the negative susceptibility in holography and its resolution.