Hydrodynamic tails and a fluctuation bound on the bulk viscosity

Abstract

We study the small frequency behavior of the bulk viscosity spectral function using stochastic fluid dynamics. We obtain a number of model independent results, including the long-time tail of the bulk stress correlation function, and the leading non-analyticity of the spectral function at small frequency. We also establish a lower bound on the bulk viscosity which is weakly dependent on assumptions regarding the range of applicability of fluid dynamics. The bound on the bulk viscosity ζ scales as ζ min ∼ (P − 2 3 E) 2 i D −2 i , where D i are the diffusion constants for energy and momentum, and P − 2 3 E, where P is the pressure and E is the energy density, is a measure of scale breaking. Applied to the cold Fermi gas near unitarity, |λ/a s | ∼ > 1 where λ is the thermal de Broglie wave length and a s is the s-wave scattering length, this bound implies that the ratio of bulk viscosity to entropy density satisfies ζ/s ∼ > 0.1h/k B . Here,h is Planck's constant and k B is Boltzmann's constant.

High-temperature expansion of the viscosity in interacting quantum gases

“…It was established in Refs. [40,41] that the bulk viscosity can also be defined in terms of the response function…”

Critical behavior of the bulk viscosity in QCD

“…The time integral of G(t) is given by the bulk viscosity ζ, and the asymptotic behavior of G(t) for large t is governed by hydrodynamic tails G(t) ∼ 1/t 3/2 [18,20,[37][38][39][40][41][42]. The tail of the critical contribution is which has an even stronger dependence on the correlation length than the static value of the bulk viscosity.…”

Bulk Viscosity and Contact Correlations in Attractive Fermi Gases

“…It is therefore important to compute the bulk viscosity theoretically for quantum gases, which moreover includes predictions for the classical gas in the high-temperature limit.The bulk viscosity is defined as the correlation function of local pressure (the trace of the stress tensor). Since it vanishes in a scale invariant system, only the scale breaking part of the pressure contributes, the so-called trace anomaly [14,23,24]. This provides a formal link between the breaking of scale invariance and bulk viscosity.…”

“…Its value is largest in the strongly coupled region of the BEC-BCS crossover [27] near unitarity, but not precisely at unitarity where is must vanish by scale invariance [3,9,28]. Furthermore, hydrodynamic fluctuations give rise to nonanalytic corrections to the bulk viscosity at small frequencies [23,29]. However, key open questions include the bulk viscosity in degenerate Fermi gases at strong interaction, the relative importance of bulk and shear viscosity, and critical scaling near the superfluid phase transition.In this work, I rewrite the bulk viscosity of the dilute Fermi gas as a correlation function of the contact density of local fermion pairs.…”

“…Contact correlation.-By conservation of energy, the Hamiltonian in (3) does not contribute to the pressure commutator (1), and the bulk viscosity is given by the correlator of the scale breaking term. The scaling violation in the trace of the stress tensor is the so-called trace anomaly [14,23]…”