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 $e^{-n^2ω_I t}$. However, nonlinear analysis gives a decay of $e^{-nω_I t}$. 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, $J_n=α_J^{\,n-1}J_1^n$, with $α_J$ fixed by the equation of state, the longitudinal viscosity, and the fundamental wavenumber. For conformal fluids, $α_J=1/(8ηk)$, 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.