Kubo formulas for thermodynamic transport coefficients

Understand

Uncharged relativistic fluids in 3+1 dimensions have three independent thermodynamic transport coefficients at second order in the derivative expansion.

• Fluids with a single global $U(1)$ current have nine, out of which seven are parity preserving.
• We derive the Kubo formulas for all nine thermodynamic transport coefficients in terms of equilibrium correlation functions of the energy-momentum tensor and the current.
• All parity-preserving coefficients can be expressed in terms of two-point functions in flat space without external sources, while the parity-violating coefficients require three-point functions.