Kubo Formulae for Second-Order Hydrodynamic Coefficients

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.

Comments: four pages, no figures. Corrects an error pointed out by Arnold, Vaman, Wu, and Wei; the general technique of the original paper is correct but the specific 2'nd order Kubo relations were wrong and are replaced here by correct versions