Consistency of field-theoretical and kinetic calculations of viscous transport coefficients for a relativistic fluid

Gabriel S Denicol,Xu-Guang Huang,Tomoi Koide,Dirk H Rischke

doi:10.1016/j.physletb.2012.01.018

Gabriel S Denicol, Xu-Guang Huang + Show 2 more

Open Access

https://doi.org/10.1016/j.physletb.2012.01.018

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Abstract

We investigate the ratios βη≡η/τπ and βζ≡ζ/τΠ, i.e., the ratios of shear, η, and bulk, ζ, viscosities to the relaxation times τπ, τΠ of the shear stress tensor and bulk viscous pressure, respectively, in the framework of causal relativistic dissipative fluid dynamics. These viscous transport coefficients are computed both in a field-theoretical and a kinetic approach based on the Boltzmann equation. Our results differ from those of the traditional Boltzmann calculation by Israel and Stewart. The new expressions for the viscous transport coefficients agree with the results obtained in the field-theoretical approach when the contributions from pair annihilation and creation (PAC) are neglected. The latter induce non-negligible corrections to the viscous transport coefficients.

Highlights

Relativistic fluid dynamics is an important model to understand various collective phenomena in astrophysics and heavy-ion collisions
The transport coefficients for causal relativistic dissipative fluid dynamics (CRDF) cannot be computed applying the methods commonly used for Navier-Stokes fluids, such as the Green-Kubo-Nakano (GKN) formula
In a leading-order perturbative calculation which should apply in the dilute limit, i.e., the regime of applicability of the kinetic approach, the field-theoretical formula gives results which are different from those of the Israel and Stewart (IS) calculation

Summary

KμKνKρ f

In order to obtain a closed set of equations, one assumes a specific form for f , f = f0 + f0(1 − af0)(e + eμKμ + eμν KμKν ), (2). Where e, eμ, and eμν constitute a set of 14 independent parameters related to the irreversible currents by matching conditions and f0 = (eβuμKμ + a)−1 is the singleparticle distribution function in local equilibrium, with a = ±1 for fermions/bosons; β ≡ 1/T is the inverse temperature. Eq (1) is decomposed into scalar, vector, and tensor parts which are interpreted as the evolution equations of the bulk viscous pressure, the particle diffusion (heat conduction) current, and the shear stress tensor, respectively. The shear stress tensor and the bulk viscous pressure are always defined by πμν = ∆μναβ. Details will be presented elsewhere [11]

The final result for the evolution equations is

Ek β