Inverse-Reynolds-Dominance approach to transient fluid dynamics

TL;DR

This paper introduces the Inverse-Reynolds-Dominance (IReD) approach to relativistic dissipative hydrodynamics, simplifying the derivation of relaxation equations by directly matching moments to dissipative quantities, and clarifies the relation to existing methods.

Contribution

The paper proposes the IReD scheme that eliminates second-order Kn terms in relaxation equations and provides a new way to determine transport coefficients and relaxation times.

Findings

Relaxation times for higher-order moments increase with order.

The IReD approach is formally equivalent to the DNMR method up to second order.

Transport coefficients are derived solely from the inverse collision matrix.

Abstract

We consider the evolution equations for the bulk viscous pressure, diffusion current and shear tensor derived within second-order relativistic dissipative hydrodynamics from kinetic theory. By matching the higher order moments directly to the dissipative quantities, all terms which are of second order in the Knudsen number Kn vanish, leaving only terms of order $\mathcal{O}(\textrm{Re}^{-1} \textrm{Kn})$ and $\mathcal{O}(\textrm{Re}^{-2})$ in the relaxation equations, where $\textrm{Re}^{-1}$ is the inverse Reynolds number. We therefore refer to this scheme as the Inverse-Reynolds-Dominance (IReD) approach. The remaining (non-vanishing) transport coefficients can be obtained exclusively in terms of the inverse of the collision matrix. This procedure fixes unambiguously the relaxation times of the dissipative quantities, which are no longer related to the eigenvalues of the inverse of the collision matrix. In particular, we find that the relaxation times corresponding to higher-order moments grow as their order increases, thereby contradicting the \textit{separation of scales} paradigm. The formal (up to second order) equivalence with the standard DNMR approach is proven and the connection between the IReD transport coefficients and the usual DNMR ones is established.