Holographic Thermodynamics Requires a Chemical Potential for Color

Understand

The thermodynamic Euler equation for high-energy states of large-$N$ gauge theories is derived from the dependence of the extensive quantities on the number of colors $N$.

• This Euler equation relates the energy of the state to the temperature, entropy, number of degrees of freedom and its chemical potential, but not to the volume or pressure.
• In the context of the gauge/gravity duality we show that the Euler equation is dual to the generalized Smarr formula for black holes in the presence of a negative cosmological constant.
• We also match the fundamental variational equation of thermodynamics to the first law of black hole mechanics, when extended to include variations of the cosmological constant and Newton's constant.