Hydrodynamics of the Chiral Dirac Spectrum

Abstract

We derive a hydrodynamical description of the eigenvalues of the chiral Dirac spectrum in the vacuum and in the large $N$ (volume) limit. The linearized hydrodynamics supports sound waves. The stochastic relaxation of the eigenvalues is captured by a hydrodynamical instanton configuration which follows from a pertinent form of Euler equation. The relaxation from a phase of localized eigenvalues and unbroken chiral symmetry to a phase of de-localized eigenvalues and broken chiral symmetry occurs over a time set by the speed of sound. We show that the time is $Δτ=πρ(0)/2βN$ with $ρ(0)$ the spectral density at zero virtuality and $β=1,2,4$ for the three Dyson ensembles that characterize QCD with different quark representations in the ergodic regime.