On the Relation of Exact Hydrodynamics to the Chapman-Enskog Series

TL;DR

This paper shows that the Chapman-Enskog series aligns with exact spectral hydrodynamics at low Knudsen numbers but diverges elsewhere, highlighting the limitations of traditional expansions versus exact spectral methods.

Contribution

It establishes the local equivalence of the Chapman-Enskog series to exact spectral hydrodynamics and demonstrates the divergence of the series outside equilibrium.

Findings

Chapman-Enskog series is locally equivalent to exact spectral closure at low Knudsen number.

The series diverges everywhere except at global equilibrium.

Exact spectral hydrodynamics are valid globally for any Knudsen number.

Abstract

We demonstrate that the Chapman-Enskog series is locally equivalent to the exact spectral closure defined on slow kinetic eigenmodes in the limit of vanishing Knudsen number. We further show that the Chapman-Enskog series diverges everywhere expect at the global equilibrium for an explicit example, while the exact spectrally closed hydrodynamics are defined globally for any Knudsen number.