Nonlinear nature of near-equilibrium viscous fluids

Abstract

We study the late-time relaxation of a neutral relativistic viscous fluid in $d+1$ dimensions. In the long-wavelength regime, linearized hydrodynamics predicts that the sound mode at momentum $nk$ decays as $e^{-n^2{\omega }_{I}t}$. However, nonlinear analysis gives a decay of $e^{-n{\omega }_{I}t}$. We derive a closed asymptotic attractor solution in which the frequency of the $n$-th harmonic locks to $n$ times the complex frequency of the fundamental mode. The amplitude envelopes for energy current $J$ obey a simple cascading relation, $J_n={\alpha }_J^{n-1}{J}_1^n$, with ${\alpha }_J$ fixed by the equation of state, the longitudinal viscosity, and the fundamental wavenumber. For conformal fluids, ${\alpha }_J=1/(8\eta k)$, in agreement with the holographic result of arXiv:2512.07242. The existence of the attractor shows that, even near equilibrium, field powers are not equivalent to amplitude order.

Comments: 19 pages