From moments of the distribution function to hydrodynamics: The non-conformal case

Abstract

We study the one-dimensional boost-invariant Boltzmann equation in the relaxation-time approximation using special moments of the distribution function for a system with a finite particle mass. The infinite hierarchy of moments can be truncated by keeping only the three lowest moments that correspond to the three independent components of the energy-momentum tensor. We show that such a three-moment truncation reproduces accurately the exact solution of the kinetic equation after a simple renormalization that takes into account the effects of the neglected higher moments. We derive second-order Israel-Stewart hydrodynamic equations from the three-moment equations, and show that, for most physically relevant initial conditions, these equations yield results comparable to those of the three-moment truncation, albeit less accurate. We attribute this feature to the fact that the structure of Israel-Stewart equations is similar to that of the three-moment truncation. In particular, the presence of the relaxation term in the Israel-Stewart equations, yields an early-time regime that mimics approximately the collisionless regime. A detailed comparison of the three-moment truncation with second-order non-conformal hydrodynamics reveals ambiguities in the definition of second-order transport coefficients. These ambiguities affect the ability of Israel-Stewart hydrodynamics to reproduce results of kinetic theory.