On the convergence of the gradient expansion in hydrodynamics

TL;DR

For the strongly coupled N=4 supersymmetric Yang-Mills plasma, the holographic duality methods are used to demonstrate that the derivative expansions have finite nonzero radii of convergence.

Abstract

Hydrodynamic excitations corresponding to sound and shear modes in fluids are characterised by gapless dispersion relations. In the hydrodynamic gradient expansion, their frequencies are represented by power series in spatial momenta. We investigate the analytic structure and convergence properties of the hydrodynamic series by studying the associated spectral curve in the space of complexified frequency and complexified spatial momentum. For the strongly coupled ${\cal N}=4$ supersymmetric Yang-Mills plasma, we use the holographic duality methods to demonstrate that the derivative expansions have finite non-zero radii of convergence. Obstruction to the convergence of hydrodynamic series arises from level-crossings in the quasinormal spectrum at complex momenta.

Figures (2)

• Figure 1: Coefficients of the expansions \ref{['eq:shear-1']} and \ref{['eq:sound-1']} in ${\cal N}=4$ SYM theory. The circles are $\ln|c_n|$ (shear mode), the squares are $\ln|a_n|$ (sound mode). Red (blue) colour indicates positive (negative) values of $c_n$ or $a_n$.
• Figure 2: Poles of the retarded two-point function of the energy-momentum tensor in the complex $\mathfrak{w}$-plane, at various values of the complexified momentum $\mathfrak{q}^2 = |\mathfrak{q}^2|e^{i \theta}$. Top row is the shear channel, bottom row is the sound channel. Large dots correspond to the location of the poles for real $\mathfrak{q}^2$ ($\theta=0$) Kovtun:2005ev. The hydrodynamic shear and sound poles are the poles closest to the real axis in the top left and bottom left panels, correspondingly. As $\theta$ increases from $0$ to $2\pi$, each pole moves counter-clockwise, following the trajectory of its colour. In the shear channel (top row), at $|\mathfrak{q}^2|=1$, each pole follows a closed orbit (top left). At $|\mathfrak{q}^2|=2.22$ (top centre), the hydrodynamic pole almost collides with the two gapped poles closest to the real axis. The actual collision would happen at the critical momentum (\ref{['eq:qw-crit-shear']}), $|\mathfrak{q}_{\rm c}^2| \approx 2.224$, with the corresponding frequencies marked by red asterisks in the figure. At $|\mathfrak{q}^2|=2.23$ (top right), the orbits of the three uppermost poles are no longer closed: the hydrodynamic pole and the two gapped poles exchange their positions cyclically as the phase $\theta$ increases from $0$ to $2\pi$. Similar behaviour is observed for the sound mode (bottom row). The dispersion relations $\mathfrak{w}_i(\mathfrak{q})$ thus have branch cuts starting at $\mathfrak{q}_{\rm c}$.