Hydrodynamic gradient expansion in linear response theory

TL;DR

This paper investigates the divergence properties of the hydrodynamic gradient expansion in relativistic fluids, revealing that divergence occurs beyond a critical momentum linked to microscopic theory singularities, independent of flow symmetry.

Contribution

It establishes the precise conditions for divergence of the hydrodynamic gradient expansion in linear response, applicable to broad microscopic theories without symmetry constraints.

Findings

Gradient expansion diverges beyond a critical momentum.

Convergence occurs when fields are supported below this critical momentum.

Critical momentum is determined by singularities in dispersion relations.

Abstract

A foundational question in relativistic fluid mechanics concerns the properties of the hydrodynamic gradient expansion at large orders. We establish the precise conditions under which this gradient expansion diverges for a broad class of microscopic theories admitting a relativistic hydrodynamic limit, in the linear regime. Our result does not rely on highly symmetric fluid flows utilized by previous studies of heavy-ion collisions and cosmology. The hydrodynamic gradient expansion diverges whenever energy density or velocity fields have support in momentum space exceeding a critical momentum, and converges otherwise. This critical momentum is an intrinsic property of the microscopic theory and is set by branch point singularities of hydrodynamic dispersion relations.