Hydrodynamics with multiple charges and holography

TL;DR

This paper explores the link between thermodynamic and dynamical instabilities in relativistic hydrodynamics with multiple conserved charges, using holographic duality to analyze instabilities in strongly coupled gauge theories.

Contribution

It establishes a theoretical connection between thermodynamic and dynamical instabilities in multi-charge relativistic fluids and demonstrates this through holographic models involving black branes.

Findings

Hydrodynamic instabilities occur only with negative diffusion coefficients.

Eigenvalues of the thermodynamic Hessian govern the onset of instabilities.

Analytic expressions for R-charge diffusion coefficients are derived.

Abstract

We establish the connection between thermodynamic and dynamical instabilities in relativistic hydrodynamics with multiple flavours of conserved U(1) charges. In theories with positive hydrodynamic entropy production, where the underlying perfect fluid has a positive speed of sound squared and satisfies the null energy condition, we show that hydrodynamic instabilities can arise only through negative diffusion coefficients associated with the U(1) charges. The onset of such instabilities is governed by the eigenvalues of the thermodynamic Hessian matrix, while the flavour-space polarisations of the unstable diffusion modes are determined by the corresponding eigenvectors. We illustrate this connection using strongly coupled N=4 supersymmetric Yang-Mills theory at finite densities of the three U(1) R-charges. In the dual holographic description, the five-dimensional STU black brane exhibits unstable quasinormal modes precisely at the onset of thermodynamic instability. We derive analytic expressions for the R-charge diffusion coefficients in several representative cases, including the configuration with three equal chemical potentials.