Kubo Formulae for Second-Order Hydrodynamic Coefficients

TL;DR

The paper derives Kubo relations for all second-order conformal hydrodynamic coefficients $\tau_{\Pi}$, $\kappa$, $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ by coupling a conformal fluid to weak metric perturbations and using the Schwinger-Keldysh formalism to express $\langle T^{\mu\nu}\rangle$ through fully retarded stress-tensor correlators. It provides explicit formulas for $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ in terms of $G_{raa}$, and shows that $\lambda_{3}$ can be computed from Euclidean correlators as $\lambda_{3} = 4 \lim_{\vec p,\vec q\to0} \partial_{p_z}\partial_{q_z} G_E^{xy,x0,y0}(p,q)$; a weak-coupling scalar-field example yields $\lambda_{3} = T^2/12$. These results place the second-order coefficients on a solid footing across coupling regimes and illuminate the thermodynamic nature of $\lambda_{3}$, differentiating it from the other coefficients which rely on dynamic (frequency-dependent) correlators. The work thus provides a practical, first-principles framework for calculating second-order transport in conformal fluids and clarifies the physical interpretation of $\lambda_{3}$.

Abstract

At second order in gradients, conformal relativistic hydrodynamics depends on the viscosity eta and on five additional "second-order" hydrodynamical coefficients tauPi, kappa, lambda1, lambda2, and lambda3. We derive Kubo relations for these coefficients, relating them to equilibrium, fully retarded 3-point correlation functions of the stress tensor. We show that the coefficient lambda3 can be evaluated directly by Euclidean means and does not in general vanish.