On the universal identity in second order hydrodynamics

TL;DR

This paper calculates the next-to-leading order corrections to second order transport coefficients in strongly coupled ${ m N}=4$ SYM theory and confirms a universal identity among these coefficients persists beyond leading order.

Contribution

It extends the understanding of second order hydrodynamics in holographic theories by computing higher-derivative corrections and verifying the universality of a key transport coefficient identity.

Findings

The identity $2 \, \eta \tau_\Pi - 4 \lambda_1 - \lambda_2=0$ holds at next-to-leading order in ${\rm N}=4$ SYM.

Corrections from curvature squared terms preserve the transport coefficient identity to linear order.

The results have implications for entropy production in strongly coupled near-equilibrium fluids.

Abstract

We compute the 't Hooft coupling correction to the infinite coupling expression for the second order transport coefficient $\lambda_2$ in ${\cal N}=4$ $SU(N_c)$ supersymmetric Yang-Mills theory at finite temperature in the limit of infinite $N_c$, which originates from the $R^4$ terms in the low energy effective action of the dual type IIB string theory. Using this result, we show that the identity involving the three second order transport coefficients, $2 \eta \tau_\Pi - 4 \lambda_1 - \lambda_2 =0$, previously shown by Haack and Yarom to hold universally in relativistic conformal field theories with string dual descriptions to leading order in supergravity approximation, holds also at next to leading order in this theory. We also compute corrections to transport coefficients in a (hypothetical) strongly interacting conformal fluid arising from the generic curvature squared terms in the corresponding dual gravity action (in particular, Gauss-Bonnet action), and show that the identity holds to linear order in the higher-derivative couplings. We discuss potential implications of these results for the near-equilibrium entropy production rate at strong coupling.