[Paper Review] The regime of applicability of Israel-Stewart hydrodynamics

David Wagner, Lorenzo Gavassino|arXiv (Cornell University)|Sep 26, 2023

Fluid Dynamics and Turbulent FlowsEngineering3 citations

TL;DR

This paper systematically evaluates the accuracy of microscopic derivations of Israel-Stewart hydrodynamics near local equilibrium using linear response theory. It ranks approaches by accuracy—Inversely Reynolds-Dominated (IReD) and Denicol-Niemi-Molnár-Rischke (DNMR) as most accurate—demonstrating that Israel-Stewart theory is falsifiable and that the relaxation time is observable, resolving long-standing debates on its physical validity beyond Navier-Stokes.

ABSTRACT

Using analytical tools from linear response theory, we systematically assess the accuracy of several microscopic derivations of Israel-Stewart hydrodynamics near local equilibrium. This allows us to "rank" the different approaches in decreasing order of accuracy as follows: Inverse Reynolds Dominance (IReD), Denicol-Niemi-Molnár-Rischke (DNMR), second-order gradient expansion, and 14-moment approximation. We find that IReD theory is far superior to Navier-Stokes, being very accurate both in the asymptotic regime (i.e., for slow processes) and in the transient regime (i.e., on timescales comparable to the relaxation time). Also, the high accuracy of DNMR is confirmed, but neglecting second-order terms in the Knudsen number, which would render the equations parabolic, introduces serious systematic errors. Finally, in most cases, the second-order gradient expansion (a.k.a. non-resummed BRSSS) is found to be more inaccurate than Navier-Stokes in the transient regime. Overall, this analysis shows that Israel-Stewart hydrodynamics is falsifiable, and the relaxation time is observable, shedding new light on the debate on the viability of transient hydrodynamics as a well-defined physical theory distinguished from Navier-Stokes.

Motivation & Objective

• To resolve the long-standing debate on whether Israel-Stewart hydrodynamics extends beyond Navier-Stokes or merely improves it.
• To assess the accuracy of various microscopic derivations of Israel-Stewart hydrodynamics near local equilibrium.
• To determine which derivation methods correctly capture both asymptotic and transient hydrodynamic regimes.
• To clarify whether the relaxation time is an observable physical parameter or merely a regulator.
• To establish a systematic framework for comparing hydrodynamic models based on their fidelity to kinetic theory.

Proposed method

• Uses linear response theory to analyze the accuracy of different hydrodynamic derivations in the regime near local equilibrium.
• Derives and compares the linearized moment equations for irreducible moments of the phase-space distribution function.
• Identifies and evaluates four distinct approaches: Inverse Reynolds Dominance (IReD), Denicol-Niemi-Molnár-Rischke (DNMR), second-order gradient expansion, and 14-moment approximation.
• Inverts the linearized collision matrices to derive relaxation time tensors and transport coefficients from kinetic theory.
• Compares the resulting equations to the standard Israel-Stewart form, assessing consistency with causality and stability.
• Employs a worldline-based reduction to map the infinite-dimensional moment hierarchy to a finite set of hydrodynamic variables.

Experimental results

Research questions

• RQ1 Which microscopic derivation of Israel-Stewart hydrodynamics most accurately captures the dynamics near local equilibrium?
• RQ2 Is the relaxation time in Israel-Stewart theory an observable physical parameter or merely a UV regulator?
• RQ3 Does the second-order gradient expansion (non-resummed BRSSS) outperform Navier-Stokes in the transient regime?
• RQ4 Can the DNMR approach maintain accuracy when second-order gradient terms are neglected for causality?
• RQ5 Does IReD theory provide a more accurate description than Navier-Stokes in both asymptotic and transient regimes?

Key findings

• IReD theory is found to be significantly more accurate than Navier-Stokes, performing well in both asymptotic and transient regimes.
• DNMR theory is confirmed as highly accurate, but neglecting second-order gradient terms introduces serious systematic errors that compromise causality.
• The second-order gradient expansion (non-resummed BRSSS) is less accurate than Navier-Stokes in the transient regime, contrary to expectations.
• The 14-moment approximation is the least accurate among the four methods evaluated, especially in capturing non-equilibrium dynamics.
• The relaxation time is shown to be observable and physically meaningful, not just a regulator, supporting the viability of transient hydrodynamics as a distinct physical theory.
• Israel-Stewart hydrodynamics is falsifiable, as its predictions can be tested against kinetic theory in controlled regimes.