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arXiv:1007.5333 · Physical Review Letters

Kubo Formulas for Second-Order Hydrodynamic Coefficients

Guy D. Moore, Kiyoumars A. Sohrabi

Published Mar 23, 2011• 85 citations
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Abstract

At second order in gradients, conformal relativistic hydrodynamics depends on the viscosity $\ensuremath{\eta}$ and on five additional ``second-order'' hydrodynamical coefficients ${\ensuremath{\tau}}_{\ensuremath{\Pi}}$, $\ensuremath{\kappa}$, ${\ensuremath{\lambda}}_{1}$, ${\ensuremath{\lambda}}_{2}$, and ${\ensuremath{\lambda}}_{3}$. We derive Kubo relations for these coefficients, relating them to equilibrium, fully retarded three-point correlation functions of the stress tensor. We show that the coefficient ${\ensuremath{\lambda}}_{3}$ can be evaluated directly by Euclidean means and does not in general vanish.

DOI
10.1103/physrevlett.106.122302

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At second order in gradients, conformal relativistic hydrodynamics depends on the viscosity \ensuremathη\ensuremath{\eta}\ensuremathη and on five additional ``second-order'' hydrodynamical coefficients \ensuremathτ\ensuremathΠ{\ensuremath{\tau}}_{\ensuremath{\Pi}}\ensuremathτ\ensuremathΠ​, \ensuremathκ\ensuremath{\kappa}\ensuremathκ, \ensuremathλ1{\ensuremath{\lambda}}_{1}\ensuremathλ1​, \ensuremathλ2{\ensuremath{\lambda}}_{2}\ensuremathλ2​, and \ensuremathλ3{\ensuremath{\lambda}}_{3}\ensuremathλ3​. We derive Kubo relations for these coefficients, relating them to equilibrium, fully retarded three-point correlation functions of the stress tensor. We show that the coefficient \ensuremathλ3{\ensuremath{\lambda}}_{3}\ensuremathλ3​ can be evaluated directly by Euclidean means and does not in general vanish.
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Motivated by recent work on Hall viscosity, we derive from first principles the Kubo formulas for the stress-stress response function at zero wave vector that can be used to define the full complex frequency-dependent viscosity tensor, both with and without a uniform magnetic field. The formulas in the existing literature are frequently incomplete, incorrect, or lack a derivation; in particular, Hall viscosity is overlooked. Our approach begins from the response to a uniform external strain field, which is an active time-dependent coordinate transformation in ddd space dimensions. These transformations form the group GL(d,R)(d,\mathbb{R})(d,R) of invertible matrices, and the infinitesimal generators are called strain generators. These enable us to express the Kubo formula in different ways, related by Ward identities; some of these make contact with the adiabatic transport approach. The importance of retaining contact terms, analogous to the diamagnetic term in the familiar Kubo formula for conductivity, is emphasized. For Galilean-invariant systems, we derive a relation between the stress response tensor and the conductivity tensor that is valid at all frequencies and in both the presence and absence of a magnetic field. In the presence of a magnetic field and at low frequency, this yields a relation between the Hall viscosity, the q2{q}^{2}q2 part of the Hall conductivity, the inverse compressibility (suitably defined), and the diverging part of the shear viscosity (if any); this relation generalizes a result found recently by others. We show that the correct value of the Hall viscosity at zero frequency can be obtained (at least in the absence of low-frequency bulk and shear viscosity) by assuming that there is an orbital spin per particle that couples to a perturbing electromagnetic field as a magnetization per particle. We study several examples as checks on our formulation. We also present formulas for the stress response that directly generalize the Berry (adiabatic) curvature expressions for zero-frequency Hall conductivity or viscosity to the full tensors at all frequencies.
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