Constructing higher-order hydrodynamics: The third order

TL;DR

This paper extends relativistic hydrodynamics to third order in the gradient expansion, classifying new transport coefficients and analyzing their physical effects in various fluid flow scenarios.

Contribution

It systematically classifies third-order transport coefficients for neutral fluids in four dimensions, providing the most general set without entropy current constraints.

Findings

Identified 20 new conformal and 68 non-conformal third-order transport coefficients.

Derived third-order corrections to wave dispersion relations and stress-energy correlations.

Applied results to $ ext{N}=4$ SYM fluid to determine specific coefficient combinations.

Abstract

Hydrodynamics can be formulated as the gradient expansion of conserved currents in terms of the fundamental fields describing the near-equilibrium fluid flow. In the relativistic case, the Navier-Stokes equations follow from the conservation of the stress-energy tensor to first order in derivatives. In this paper, we go beyond the presently understood second-order hydrodynamics and discuss the systematisation of obtaining the hydrodynamic expansion to an arbitrarily high order. As an example of the algorithm that we present, we fully classify the gradient expansion at third order for neutral fluids in four dimensions, thus finding the most general next-to-leading-order corrections to the relativistic Navier-Stokes equations in curved space-time. In doing so, we list $20$ new transport coefficient candidates in the conformal and $68$ in the non-conformal case. As we do not consider any constraints that could potentially arise from the local entropy current analysis, this is the maximal possible set of neutral third-order transport coefficients. To investigate the physical implications of these new transport coefficients, we obtain the third-order corrections to the linear dispersion relations that describe the propagation of diffusion and sound waves in relativistic fluids. We also compute the corrections to the scalar (spin-$2$) two-point correlation function of the third-order stress-energy tensor. Furthermore, as an example of a non-linear hydrodynamic flow, we calculate the third-order corrections to the energy density of a boost-invariant Bjorken flow. Finally, we apply our field theoretic results to the $\mathcal{N}=4$ supersymmetric Yang-Mills fluid at infinite 't Hooft coupling and infinite number of colours to find the values of five new linear combinations of the conformal transport coefficients.