The Spatial Hydrodynamic Attractor: Resurgence of the Gradient Expansion

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.

Figures (2)

• Figure 1: Absolute values of the exact CE coefficients $|a_{2n}|$ from Eq. \ref{['eq:a2n_exact']} (black circles, left axis) and ratios $r_n = |a_{2(n+1)}/a_{2n}|$ (blue squares, right axis). The convex shape of $|a_{2n}|$ on a log scale signals factorial divergence, while the asymptotically linear growth $r_n \sim 2n$ implies zero radius of convergence. This factorial scaling is the spatial-gradient analog of the temporal divergence figure in Ref. Heller:2015; crucially, the spatial series is strictly Borel summable.
• Figure 2: Hydrodynamic dispersion relation $\hat{\omega}(k)$. Solid colored curves show the spectral polynomial branches $P_n(\hat{\omega},k^2)=0$ for $n=1,2,20,50$, converging to the non-perturbative attractor with increasing truncation order; faded segments beyond the critical wavenumber $k_c$ (filled circles) indicate the unphysical region past the fold bifurcation, with $k_c \approx 0.47, 0.58, 0.94, 1.03$ for $n=1,2,20,50$ respectively. The diagonal Padé approximant of order $[14/14]$, constructed from the first 30 Borel-transformed CE coefficients (red), exactly reconstructs the attractor. Finite-order CE truncations CE$^{(2)}$ (dashed) and CE$^{(4)}$ (dash-dotted) are both divergent.