Hydrodynamics of the Chiral Dirac Spectrum

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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.