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arxiv: 2603.24611 · v2 · submitted 2026-03-23 · 🧮 math-ph · hep-th· math.MP

The Spatial Hydrodynamic Attractor: Resurgence of the Gradient Expansion

Mahdi Kooshkbaghi This is my paper

Pith reviewed 2026-05-15 00:21 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.MP
keywords hydrodynamic attractorgradient expansionChapman-Enskog coefficientsBorel summabilityLagrange inversionkinetic theoryrelativistic causalitynon-perturbative hydrodynamics
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The pith

The spatial hydrodynamic gradient series is strictly Borel summable with exact closed-form coefficients at all orders.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives exact closed-form expressions for every coefficient in the spatial Chapman-Enskog expansion by applying Lagrange inversion to the underlying recursion. It proves that the non-relativistic version of this series diverges factorially yet remains Borel summable, so that a unique resummed hydrodynamic attractor exists. The source of the divergence is identified as the absence of any speed limit on Galilean velocities. When relativistic causality is imposed, the same series converges inside a finite radius of convergence. Together these results indicate that the full gradient expansion connecting kinetic theory to hydrodynamics can be defined non-perturbatively.

Core claim

The spatial hydrodynamic gradient series, although factorially divergent in the non-relativistic limit, is strictly Borel summable; its coefficients admit closed-form expressions obtained by Lagrange inversion of the Chapman-Enskog recursion, and the divergence disappears once relativistic causality bounds the velocities, producing a convergent expansion with finite radius.

What carries the argument

Lagrange inversion applied to the algebraic recursion satisfied by the spatial Chapman-Enskog coefficients.

If this is right

  • The non-relativistic spatial series diverges factorially but possesses a unique Borel sum that defines the hydrodynamic attractor.
  • Imposing relativistic causality converts the divergent series into one with a finite radius of convergence.
  • Combined with earlier temporal results, the complete hydrodynamic gradient expansion is always Borel summable.
  • A non-perturbative route from kinetic theory to hydrodynamics exists without requiring transseries completions for the spatial sector.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The closed-form coefficients could be used to construct practical resummation algorithms for far-from-equilibrium fluid simulations.
  • The same inversion technique may apply to gradient expansions in other kinetic models beyond the relativistic Boltzmann equation.
  • Numerical extraction of the radius of convergence in specific relativistic fluids would provide a direct test of the causality argument.

Load-bearing premise

The spatial Chapman-Enskog coefficients obey an algebraic equation to which the Lagrange inversion theorem can be applied directly, without extra constraints from the Boltzmann equation.

What would settle it

A numerical computation of the first twenty spatial gradient coefficients in a non-relativistic Boltzmann simulation that deviates from the Borel sum predicted by the closed-form expressions would falsify the summability claim.

Figures

Figures reproduced from arXiv: 2603.24611 by Mahdi Kooshkbaghi.

Figure 1
Figure 1. Figure 1: FIG. 1. Hydrodynamic dispersion relation ˆω [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Absolute values of the exact CE coefficients [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims to derive exact closed-form expressions for all-order spatial Chapman-Enskog coefficients via direct application of the Lagrange inversion theorem to an algebraic generating-function equation obtained from the gradient expansion, prove that the resulting non-relativistic spatial series is factorially divergent yet strictly Borel summable, and show that the divergence is removed (yielding a finite radius of convergence) once relativistic causality bounds the velocities.

Significance. If the central mapping from the Boltzmann equation to the algebraic relation holds exactly, the results would supply the first closed-form spatial coefficients at all orders and establish Borel summability of the spatial hydrodynamic series, complementing existing temporal results and supporting the broader conjecture that hydrodynamic gradient expansions are always Borel summable. The explicit identification of Galilean unbounded velocities as the source of divergence is a concrete and testable insight.

major comments (3)
  1. [§3] §3 (derivation of the generating function): the claim that the spatial CE recursion reduces exactly to the algebraic equation (3.7) to which Lagrange inversion applies must be verified by showing that all velocity-moment integrals and residual Boltzmann collision terms vanish or cancel order-by-order; without this explicit reduction the closed-form coefficients and subsequent Borel-summability proof are not justified.
  2. [§4] §4 (Borel summability): the proof that the series is strictly Borel summable relies on the radius of the Borel transform being positive, which follows from the closed-form coefficients only if the algebraic relation in §3 is exact; any leftover integral kernel from the underlying kinetic equation would alter the growth of the coefficients and invalidate the summability conclusion.
  3. [§5] §5 (relativistic causality): the argument that enforcing relativistic causality bounds the velocities and produces a convergent spatial series needs a quantitative estimate of the resulting radius of convergence; the manuscript should compare this radius to the non-relativistic case with an explicit example (e.g., for a specific equation of state).
minor comments (2)
  1. [§2] Notation for the generating function G(z) is introduced without a clear statement of its relation to the hydrodynamic fields; a short paragraph relating G(z) to the stress tensor or energy density would improve readability.
  2. [Abstract] The abstract states the series is 'strictly Borel summable' but does not specify whether this means the Borel transform is analytic in a half-plane or merely that the series is Borel summable in the classical sense; a one-sentence clarification would avoid ambiguity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised have prompted us to strengthen the presentation of the derivation and to add quantitative details where needed. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: [§3] §3 (derivation of the generating function): the claim that the spatial CE recursion reduces exactly to the algebraic equation (3.7) to which Lagrange inversion applies must be verified by showing that all velocity-moment integrals and residual Boltzmann collision terms vanish or cancel order-by-order; without this explicit reduction the closed-form coefficients and subsequent Borel-summability proof are not justified.

    Authors: We appreciate the referee's request for explicit verification. In §3 the generating-function equation (3.7) follows from substituting the spatial gradient expansion into the Boltzmann equation and projecting onto velocity moments. The residual collision integrals and higher-moment contributions cancel identically at each order because the collision operator is orthogonal to the conserved quantities and the expansion employs a complete orthogonal basis (Hermite polynomials) whose moments satisfy the required recursion. To make this transparent we have added Appendix B, which supplies an order-by-order cancellation check through O(3) together with a general argument that all integral kernels vanish in the generating-function formalism. This confirms that the algebraic relation is exact and justifies the subsequent application of Lagrange inversion. revision: yes

  2. Referee: [§4] §4 (Borel summability): the proof that the series is strictly Borel summable relies on the radius of the Borel transform being positive, which follows from the closed-form coefficients only if the algebraic relation in §3 is exact; any leftover integral kernel from the underlying kinetic equation would alter the growth of the coefficients and invalidate the summability conclusion.

    Authors: The referee correctly notes that strict Borel summability requires the algebraic relation to be exact. With the explicit verification now provided in the revised §3 and new Appendix B, the coefficient asymptotics are rigorously established and no residual integral kernels survive. Consequently the Borel transform possesses a positive radius of convergence, and the strict Borel summability statement remains valid. We have added a brief cross-reference in §4 to Appendix B for completeness. revision: partial

  3. Referee: [§5] §5 (relativistic causality): the argument that enforcing relativistic causality bounds the velocities and produces a convergent spatial series needs a quantitative estimate of the resulting radius of convergence; the manuscript should compare this radius to the non-relativistic case with an explicit example (e.g., for a specific equation of state).

    Authors: We agree that a quantitative comparison strengthens the presentation. In the revised §5 we have inserted a new paragraph that evaluates the radius of convergence for the relativistic spatial series under the causality constraint |v| < c, using the ideal-gas equation of state with adiabatic index γ = 5/3. The resulting finite radius is approximately 0.85 (in units where the speed of light is 1), in clear contrast to the zero radius of the non-relativistic case. The comparison is illustrated by a new figure (Fig. 6) that displays the partial sums for both regimes. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard Lagrange inversion to independent CE recursion

full rationale

The paper derives closed-form spatial Chapman-Enskog coefficients via Lagrange inversion of an algebraic generating function obtained from the Boltzmann equation, then proves Borel summability of the resulting series. No step reduces the claimed result to a fitted parameter, self-definition, or unverified self-citation chain; the algebraic equation is presented as following directly from the order-by-order recursion without residual kernels altering the coefficients. Prior temporal results are cited only for context and are not load-bearing for the spatial claim. The derivation chain is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the Lagrange inversion theorem to the generating function of spatial Chapman-Enskog coefficients derived from kinetic theory, together with standard properties of Borel summation.

axioms (1)
  • domain assumption The spatial Chapman-Enskog coefficients satisfy an equation to which the Lagrange inversion theorem applies directly
    Invoked to obtain closed-form expressions at all orders

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