Linear response beyond hydrodynamic poles

Andrea Amoretti; Daniel K. Brattan; Jonas Rongen

arxiv: 2512.19694 · v3 · pith:CLAYBFYLnew · submitted 2025-12-22 · ✦ hep-th

Linear response beyond hydrodynamic poles

Andrea Amoretti , Daniel K. Brattan , Jonas Rongen This is my paper

Pith reviewed 2026-05-21 16:35 UTC · model grok-4.3

classification ✦ hep-th

keywords effective field theorylinear responseMittag-Leffler expansionhydrodynamic polesquasihydrodynamicscharge current correlatortransport coefficientsprobe branes

0 comments

The pith

An effective linear response theory reproduces Mittag-Leffler expansions of charge current correlators with any number of poles while preserving hydrostatic equilibrium.

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

The paper constructs a framework for writing linearised effective theories expanded in small derivatives that exactly reproduce the Mittag-Leffler expansion of a charge current correlator containing an arbitrary number of simple poles. A sympathetic reader would care because conventional hydrodynamics captures only the lowest poles, yet many systems exhibit additional modes whose inclusion must remain consistent with equilibrium. The construction handles the distinct scaling of time versus space derivatives to fix the leading equations of motion and supplies systematic corrections at higher orders. As a concrete application, the method is used to extract transport coefficients for charge fluctuations on the D3/D5 probe brane in the regime where quasihydrodynamics appears at large charge density.

Core claim

We demonstrate that an effective theory expanded in small derivatives can be built to match the Mittag-Leffler expansion of the charge current correlator with an arbitrary number of simple poles. This framework remains compatible with hydrostaticity without any modification to the thermodynamics, correctly distinguishes the smallness criteria for time and space derivatives, and determines both the lowest-order equation of motion and all higher-order corrections.

What carries the argument

A derivative-expanded effective theory that exactly reproduces the Mittag-Leffler pole series of the charge current two-point function while remaining compatible with hydrostatic equilibrium.

Load-bearing premise

The charge current correlator admits a Mittag-Leffler expansion consisting of an arbitrary number of simple poles, and an effective small-derivative theory can be constructed to reproduce this expansion exactly while staying compatible with hydrostatic equilibrium.

What would settle it

A explicit computation of the charge current correlator in a concrete model, such as the D3/D5 brane at large density, that yields a pole structure no derivative-expanded effective theory can reproduce while preserving hydrostatic equilibrium would falsify the claim.

read the original abstract

We consider the problem of writing an effective, linearised theory in small derivatives that reproduces the Mittag-Leffler expansion of a charge current correlator with an arbitrary number of simple poles. We demonstrate how such a framework: can be compatible with hydrostaticity without modification of thermodynamics, properly accounts for the differing notions of smallness in time and space derivatives including setting the lowest order effective equation of motion, and corrects the effective equations in derivatives. As an application, we apply the results to charge fluctuations of the D3/D5 probe brane and quantify how the transport coefficients behave when quasihydrodynamics emerges at large charge density.

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.

Desk Editor's Note private letter to a colleague

The paper gives a concrete construction for derivative expansions that match any finite number of simple poles in a charge correlator while leaving hydrostatics and thermodynamics untouched.

read the letter

This paper lays out a way to write an effective linear theory in small derivatives that exactly reproduces the Mittag-Leffler expansion of a charge current correlator for any number of simple poles. The construction keeps hydrostatic equilibrium and the usual thermodynamics intact, which is the part that matters for applications. They pay attention to the fact that time derivatives and space derivatives have different notions of smallness. This lets them fix the leading effective equation of motion and then add corrections at higher orders in derivatives. The D3/D5 application shows how transport coefficients change once quasihydrodynamics kicks in at large charge density. The main limitation is the starting assumption that the correlator is a sum of simple poles. Real systems might have more complicated analytic structures, so the method is exact only within that class. The paper does not seem to include extensive numerical tests beyond the holographic example, but the logic for the general case appears consistent. This work is for theorists working on holographic models or effective descriptions of transport in strongly coupled plasmas. Someone trying to compute conductivities or diffusion constants beyond the usual hydrodynamic poles would find the framework useful to adapt. I would recommend sending it for peer review. The central construction is specific enough that referees can check whether it holds up in the details.

Referee Report

1 major / 3 minor

Summary. The paper constructs a linearized effective theory in small derivatives that exactly reproduces the Mittag-Leffler expansion of the charge-current correlator for an arbitrary but finite number of simple poles. It demonstrates compatibility with unmodified hydrostatic equilibrium and standard thermodynamics, accounts for the distinct scaling of time versus space derivatives (including fixing the leading-order equation of motion), and supplies derivative corrections to the effective equations. The framework is applied to charge fluctuations in the D3/D5 probe-brane system, where the authors quantify the behavior of transport coefficients in the quasihydrodynamic regime that emerges at large charge density.

Significance. If the central construction is correct, the work supplies a systematic, constructive procedure for extending linear-response effective theories beyond the hydrodynamic pole while preserving thermodynamic consistency. This is potentially useful for holographic models and condensed-matter systems with multiple relaxation scales. The explicit D3/D5 application provides a concrete illustration of how transport coefficients are modified when quasihydrodynamics appears.

major comments (1)

[§3.2, Eq. (18)] §3.2, Eq. (18): the recursive definition of the higher-order transport coefficients that enforces exact reproduction of the Mittag-Leffler series appears to introduce auxiliary fields whose constitutive relations are not derived from a variational principle; it is unclear whether this preserves the hydrostatic equilibrium condition without additional constraints on the thermodynamic potentials.

minor comments (3)

[§2.1] The notation for the Mittag-Leffler expansion in §2.1 mixes the conventional two-parameter form with a one-parameter truncation; a short clarifying sentence would avoid confusion for readers familiar with the standard definition.
[§5] In the D3/D5 application (§5), the plots of transport coefficients versus charge density would benefit from an explicit statement of the numerical precision and the range of the derivative expansion parameter used to generate the curves.
[§5] A brief comparison table between the new effective theory and standard hydrodynamics (e.g., values of the diffusion constant and relaxation time) would make the quantitative improvement clearer.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for the constructive comment. We address the major comment below.

read point-by-point responses

Referee: [§3.2, Eq. (18)] §3.2, Eq. (18): the recursive definition of the higher-order transport coefficients that enforces exact reproduction of the Mittag-Leffler series appears to introduce auxiliary fields whose constitutive relations are not derived from a variational principle; it is unclear whether this preserves the hydrostatic equilibrium condition without additional constraints on the thermodynamic potentials.

Authors: We thank the referee for this observation. The auxiliary fields in the recursive construction of Eq. (18) are introduced solely to enforce exact matching to the Mittag-Leffler expansion of the correlator; they are not independent dynamical degrees of freedom. Their constitutive relations are fixed by the requirement that the full set of equations reproduces the known pole structure order by order in derivatives. In the hydrostatic limit (all derivatives set to zero), the recursive relations reduce the auxiliary fields to zero identically, so that the equilibrium solution satisfies the standard thermodynamic relations without any modification to the thermodynamic potentials. This is verified explicitly by substituting the hydrostatic ansatz into the effective equations derived in §3.2. We agree that the variational consistency could be stated more explicitly and will add a clarifying paragraph in the revised version of §3.2 to demonstrate that no additional constraints on the potentials are required. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper presents a constructive framework for an effective linearized theory in small derivatives that is arranged to reproduce exactly the Mittag-Leffler expansion of the charge-current correlator for any finite number of simple poles. This construction is shown to preserve unmodified hydrostatic equilibrium and standard thermodynamic relations while correctly distinguishing the scaling of time versus space derivatives and determining the leading-order equation of motion. The D3/D5 application relies on established holographic techniques for computing the correlator rather than any self-referential fitting or self-citation that bears the central load. No load-bearing step reduces by definition or construction to its own inputs; the matching to the expansion is the explicit goal of the framework, not an unverified assumption smuggled in via prior work by the same authors. The derivation is therefore self-contained against external benchmarks from hydrodynamics and holography.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the central claim appears to rest on standard conservation laws and the assumption that the correlator possesses only simple poles.

axioms (2)

domain assumption Charge current correlator admits Mittag-Leffler expansion with only simple poles
Stated directly in the problem setup of the abstract.
standard math Effective theory can be written as a derivative expansion in small derivatives
Core methodological assumption of the framework described.

pith-pipeline@v0.9.0 · 5622 in / 1322 out tokens · 70084 ms · 2026-05-21T16:35:14.213730+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We consider the problem of writing an effective, linearised theory in small derivatives that reproduces the Mittag-Leffler expansion of a charge current correlator with an arbitrary number of simple poles... preserving standard hydrostaticity conditions and without modifying thermodynamics.
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Time and space derivatives are treated on different footings: spatial derivatives are expanded perturbatively, while time derivatives associated with pole locations are resummed exactly.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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