The Spatial Hydrodynamic Attractor: Resurgence of the Gradient Expansion

TL;DR

This paper demonstrates that the spatial hydrodynamic gradient series is Borel summable and convergent under relativistic causality, revealing a non-perturbative connection between kinetic theory and hydrodynamics.

Contribution

It derives exact Chapman--Enskog coefficients at all orders and proves the Borel summability and convergence of the spatial gradient expansion.

Findings

Spatial gradient series is factorially divergent but Borel summable.

Relativistic causality enforces convergence with finite radius.

Hydrodynamic gradient expansion is always Borel summable.

Abstract

Far-from-equilibrium kinetic systems collapse onto a hydrodynamic attractor, traditionally approximated by a gradient expansion. While temporal gradient series are non-Borel summable and require transseries completions, the analytic structure of the spatial expansion has remained elusive. Here, we derive exact closed-form Chapman--Enskog coefficients at all orders via Lagrange inversion and prove that the non-relativistic spatial gradient series, though factorially divergent, is strictly Borel summable. Furthermore, we show that this divergence originates from unbounded Galilean velocities; enforcing relativistic causality yields a convergent spatial hydrodynamic expansion with finite radius. Together with prior temporal results, our findings suggest that the hydrodynamic gradient expansion is always Borel summable, pointing to a non-perturbative route from kinetic theory to hydrodynamics.