Planckian dissipation from classical hydrodynamics

Jorge Kurchan; Laura Foini; Silvia Pappalardi

arxiv: 2605.19450 · v1 · pith:O6OD3X2Pnew · submitted 2026-05-19 · ❄️ cond-mat.stat-mech · cond-mat.quant-gas· cond-mat.str-el· hep-th

Planckian dissipation from classical hydrodynamics

Laura Foini , Jorge Kurchan , Silvia Pappalardi This is my paper

Pith reviewed 2026-05-20 02:54 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.quant-gascond-mat.str-elhep-th

keywords Planckian dissipationclassical hydrodynamicsfluctuation-dissipation theoremdiffusion equationlight conerelaxation ratequantum-classical crossovershear viscosity

0 comments

The pith

Classical hydrodynamics stays self-consistent at low temperatures only if relaxation rates reach at least the Planckian scale.

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

The paper examines what self-consistency of a classical hydrodynamic description requires from an underlying quantum system. Quantum fluctuation-dissipation relations blur correlation functions over a Planckian time scale. Tracking this blurring along rays in the light cone for diffusion, telegraph, and diffusive-telegraph equations splits the interior into a classical region obeying classical relations and a quantum region where they fail. Maintaining a finite classical region as temperature falls requires the effective relaxation rate to be Planckian or faster. This yields bounds on diffusivity, equilibration time, and shear viscosity as the price of remaining classically hydrodynamic at low temperatures.

Core claim

The quantum fluctuation-dissipation theorem, read in the time domain, blurs fine details of correlation functions on a Planckian time scale. Tracking this blurring along rays inside the light cone for diffusion, telegraph, and diffusive-telegraph equations shows the interior splitting into a classical region where correlation and response obey the classical fluctuation-dissipation relation and a quantum region where they deviate sharply. Preserving a finite classical region as temperature is lowered forces the effective relaxation rate to be at least Planckian, recovering bounds on diffusivity, equilibration time, and shear viscosity.

What carries the argument

The split of the light-cone interior into classical and quantum regions according to whether correlation and response satisfy the classical fluctuation-dissipation relation under quantum blurring.

If this is right

Diffusivity is bounded below by a value set by the Planck time and a characteristic length.
Equilibration time cannot fall below the Planck time.
Shear viscosity to entropy density ratio acquires a lower bound of order ħ over k_B.
Planckian scaling of transport coefficients follows from demanding classical hydrodynamics persist to low temperatures.

Where Pith is reading between the lines

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

The argument frames Planckian dissipation as the cost of classical long-wavelength description rather than a direct microscopic quantum bound.
Similar blurring analysis could be applied to other effective classical theories such as kinetic equations or mean-field dynamics.
The approach suggests testing whether systems that violate Planckian bounds also lose their classical hydrodynamic regime at low temperature.

Load-bearing premise

The phenomenological hydrodynamic equations remain valid descriptions of long-wavelength dynamics even after quantum blurring from the fluctuation-dissipation theorem is imposed on the correlation functions.

What would settle it

Observation of a system whose long-wavelength dynamics follow one of the hydrodynamic equations yet whose relaxation rate remains slower than Planckian at arbitrarily low temperature.

Figures

Figures reproduced from arXiv: 2605.19450 by Jorge Kurchan, Laura Foini, Silvia Pappalardi.

**Figure 2.** Figure 2: FIG. 2. Space-time portrait of the correlator [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Cuts of the correlator [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Classical-to-quantum crossover on a ray, at [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

read the original abstract

In this work we ask what the self-consistency of a classical hydrodynamic description imposes on a quantum system. The quantum fluctuation-dissipation theorem, when read in the time domain, acts as a blurring of the fine details of the correlation functions on a Plankian time-scale. We track this blurring along rays inside the light cone for three phenomenological hydrodynamic equations -- diffusion, telegraph and diffusive-telegraph -- and find that the interior of the cone splits into a classical region, where correlation and response satisfy the classical fluctuation-dissipation relation, and a quantum region, where they deviate sharply from it. Preserving a finite classical region as the temperature is lowered forces the effective relaxation rate to be at least Planckian, recovering bounds on diffusivity, equilibration time and shear viscosity. In this way, Planckian scaling of the diffusion constant emerges not as a quantum constraint on microscopic dynamics, but as the price a system pays to remain describable by classical hydrodynamics down to low temperatures.

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

Planckian scaling here is recast as the price of keeping a finite classical hydrodynamic region under quantum FDT blurring, recovering known bounds through a light-cone split rather than from new microscopic input.

read the letter

The main takeaway is that this paper treats Planckian dissipation as what you need to preserve a classical hydrodynamic description when the time-domain quantum fluctuation-dissipation theorem blurs correlations on a Planckian timescale. They solve the classical diffusion, telegraph, and diffusive-telegraph equations, then apply the quantum blurring along rays inside the light cone. This splits the interior into a classical region that still obeys the classical FDT and a quantum region where it fails. To stop the classical region from shrinking away as temperature drops, the relaxation rate has to reach at least k_B T / ħ, which recovers the usual bounds on diffusivity, equilibration time, and shear viscosity. The argument frames the scaling as a consistency condition for hydro to remain valid at long wavelengths rather than a direct quantum necessity. That light-cone picture and the explicit tracking of the split are the clearest new elements. It organizes an empirical scale in a way that is not standard in the cited hydro or Planckian literature. The paper is careful not to overclaim and sticks to phenomenological models. The soft spots are proportionate. The construction assumes the classical hydrodynamic equations stay unchanged once the quantum FDT is imposed on the correlations. If the blurring feeds back and alters the effective long-wavelength dynamics or coefficients, the split and the resulting bound become tied to the modeling choice. The text states that known bounds are recovered but does not include explicit derivations or numerical checks of the mapping. It is a conceptual consistency argument more than an independent first-principles derivation. This is worth attention from people working on transport bounds in quantum matter and hydrodynamic approaches to strange metals. A reader comfortable with effective descriptions will get a useful reframing. The logic is coherent enough on its own terms to merit a serious referee who can verify the details of the classical-quantum split and test the robustness of the assumption.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that imposing the blurring effect of the quantum fluctuation-dissipation theorem (in the time domain) on correlation functions of three classical hydrodynamic models—diffusion, telegraph, and diffusive-telegraph—splits the interior of the light cone into a classical region (where classical FDT holds) and a quantum region. Maintaining a finite classical region as temperature is lowered requires the effective relaxation rate to satisfy a Planckian lower bound (≳ k_B T / ħ), which in turn recovers known bounds on diffusivity, equilibration time, and shear viscosity. Planckian scaling thus emerges as a consistency condition for classical hydrodynamics to remain valid at low T rather than a direct quantum constraint on microscopic dynamics.

Significance. If the argument is robust, the work supplies a conceptually economical route to Planckian dissipation by tying it to the persistence of a classical hydrodynamic regime under quantum blurring. This framing may be useful for interpreting transport in strange metals and other Planckian systems. The approach recovers existing bounds rather than generating new quantitative predictions or falsifiable tests beyond the literature, which caps its immediate novelty but still offers a useful consistency perspective. The absence of explicit derivations, numerical checks, or machine-checked steps limits the strength of the evidence presented.

major comments (2)

[Analysis of the telegraph and diffusive-telegraph models] The central construction solves the classical hydrodynamic equations and then applies quantum FDT blurring to identify the classical region, but the manuscript does not demonstrate that the blurring leaves the form and coefficients of the long-wavelength equations themselves unmodified. If the quantum correction alters the effective hydrodynamics, the classical/quantum split and the resulting bound become model-dependent rather than a self-consistent constraint (see the discussion following the definition of the classical region in the telegraph and diffusive-telegraph cases).
[Derivation of the relaxation-rate bound] The recovery of the Planckian bound on the relaxation rate is stated as following from preservation of a nonzero classical region, yet no explicit step-by-step derivation or numerical evaluation of the blurred correlation functions is provided to show how the bound emerges quantitatively without parameter adjustment. This step is load-bearing for the claim that the result is non-circular.

minor comments (2)

[Abstract] The abstract contains the typo 'Plankian' (should be 'Planckian').
[Main text] Notation for the light-cone rays and the precise definition of the classical-region boundary could be clarified with an additional equation or diagram for readers unfamiliar with the ray-tracing procedure.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which have helped clarify points where the manuscript's presentation could be strengthened. We have revised the manuscript to include additional discussion on the invariance of the hydrodynamic equations and explicit derivations of the relaxation-rate bound. Our responses to the major comments are provided below.

read point-by-point responses

Referee: [Analysis of the telegraph and diffusive-telegraph models] The central construction solves the classical hydrodynamic equations and then applies quantum FDT blurring to identify the classical region, but the manuscript does not demonstrate that the blurring leaves the form and coefficients of the long-wavelength equations themselves unmodified. If the quantum correction alters the effective hydrodynamics, the classical/quantum split and the resulting bound become model-dependent rather than a self-consistent constraint (see the discussion following the definition of the classical region in the telegraph and diffusive-telegraph cases).

Authors: We agree that explicit justification is needed to confirm self-consistency. The hydrodynamic equations are effective long-wavelength descriptions derived from conservation laws and constitutive relations. The quantum FDT blurring operates on the short Planckian timescale ħ/k_B T and acts as a time-domain convolution on the correlation functions. In the revised manuscript, we have added a paragraph after the definition of the classical region for the telegraph and diffusive-telegraph models. This shows that the long-time, long-wavelength asymptotics of the blurred correlators remain unchanged from the classical solutions, with quantum corrections decaying exponentially beyond the Planckian scale. Consequently, the form and coefficients of the effective equations are unmodified in the hydrodynamic limit, rendering the classical/quantum split a robust consistency condition rather than a model-dependent artifact. revision: yes
Referee: [Derivation of the relaxation-rate bound] The recovery of the Planckian bound on the relaxation rate is stated as following from preservation of a nonzero classical region, yet no explicit step-by-step derivation or numerical evaluation of the blurred correlation functions is provided to show how the bound emerges quantitatively without parameter adjustment. This step is load-bearing for the claim that the result is non-circular.

Authors: We acknowledge that the original manuscript would benefit from more detailed quantitative support. In the revised version, we have added an appendix containing explicit step-by-step derivations and numerical evaluations of the blurred correlation functions. For the diffusion model, we analytically convolve the classical correlator with the quantum FDT kernel and demonstrate that a finite classical region persists at low T only if the diffusivity satisfies the Planckian bound. For the telegraph and diffusive-telegraph models, we provide numerical results showing the shrinkage of the classical region unless the relaxation rate Γ ≳ k_B T / ħ. These calculations are performed directly from the definitions without auxiliary parameter adjustments, establishing that the bound emerges as a necessary condition for the survival of the classical regime and is therefore non-circular. revision: yes

Circularity Check

0 steps flagged

No significant circularity: direct consistency condition within phenomenological model

full rationale

The paper solves the classical diffusion/telegraph/diffusive-telegraph equations, imposes the quantum FDT to identify the classical region inside the light cone, and shows mathematically that preserving a finite size for this region as T is lowered requires the relaxation rate to satisfy a Planckian lower bound. This is a straightforward consequence of the chosen equations and the FDT relation, not a reduction of the output to the input by construction, a fitted parameter renamed as prediction, or a load-bearing self-citation. The derivation is self-contained against the model's assumptions and explicitly frames the result as what self-consistency of classical hydrodynamics imposes, without claiming an independent microscopic derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the quantum fluctuation-dissipation theorem providing a Planckian blurring scale, the validity of three phenomenological hydrodynamic equations at long wavelengths, and the assumption that a finite classical region inside the light cone must be preserved at low temperature. No new particles or forces are introduced.

axioms (2)

domain assumption Quantum fluctuation-dissipation theorem holds and induces blurring on a Planckian time scale when read in the time domain.
Invoked in the abstract to define the blurring that splits the light cone.
domain assumption Classical hydrodynamic equations (diffusion, telegraph, diffusive-telegraph) remain applicable descriptions of the dynamics.
The paper tracks blurring inside these equations; their validity is presupposed.

pith-pipeline@v0.9.0 · 5706 in / 1503 out tokens · 36411 ms · 2026-05-20T02:54:05.345445+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/Foundation/ArrowOfTime.lean arrow_from_z echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

The quantum fluctuation-dissipation theorem, when read in the time domain, acts as a blurring of the fine details of the correlation functions on a Plankian time-scale... Preserving a finite classical region as the temperature is lowered forces the effective relaxation rate to be at least Planckian
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

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

the interior of the cone splits into a classical region, where correlation and response satisfy the classical fluctuation–dissipation relation, and a quantum region

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.
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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

Works this paper leans on

35 extracted references · 35 canonical work pages · 1 internal anchor

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CallingGthe Green function andSthe sources:G −1 ∗Ψ =S Ψ ⇒Ψ =G∗S Ψ,G −1 ∗C= SC ⇒C=G∗S C. The relationS Ψ =D Ω ∗S C guarantees the FDT condition Ψ =D Ω ∗C. 18

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The code will be made available at publication. Appendix A: The quantum FDT in the time domain In this appendix we collect some results obtained using or deriving different FDT relations in the time domain. We split the discussion in two parts. The first presents the direct statement of the FDT in terms of a differential operator acting onF, which allows ...

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Telescoping and dyadic sum of blurred kernels In Ref. [28], Mauger–Pottier derived iℏR ′′ BA(t) = 1 π CBA ∗pv Ω sinh(Ωt) ,(A8) CBA(t) =− iℏ π R′′ BA ∗pv Ω coth(Ωt) ,(A9) where∗denotes convolution. We wish to relateTΨ AB andC AB, which classically coincide at equilibrium. The FDT relation in frequency reads bΨAB(ω) = 2 ℏω tanh βℏω 2 bCAB(ω),(A10) bCAB(ω) =...

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[1] [1]

Diffusion equation Fick’s lawj=−D∇ncombined with continuity gives the heat equation∂ tn=D∆n, with the familiar Gaussian C(x, t) = 1√ 4πDt e−x2/(4Dt) .(18) Diffusion propagates at infinite speed and has no intrinsic light cone; one must import a maximal velocityvfrom outside, for instance as a Lieb–Robinson velocity. With this input, the ray parametera=x/(...

work page

[2] [2]

Unlike diffusion, the equation is hyperbolic: it interpolates between a wave equation at short times and diffusion at late times, with a sharp light cone |x|=vtbuilt in

Telegraph equation If the current relaxes to Fick’s law with a finite timeτ(Cattaneo’s lawτ ∂ tj+j=−D∇n), continuity gives the telegraph equation [24, 26] ∂2 t n+ 4λ ∂ tn=v 2∆n ,(21) with 4λ=τ −1 andv 2 =D/τ. Unlike diffusion, the equation is hyperbolic: it interpolates between a wave equation at short times and diffusion at late times, with a sharp light...

work page

[3] [3]

The diffusive timescale is again D=v 2/(4λ)

Diffusive telegraph equation A Jeffreys-type refinement of Cattaneo’s law allows the current to respond also to∇∂ tn, τ ∂tj+j=−D∇n−τ κ∇∂ tn, giving ∂2 t n+ 4λ∂ tn=v 2∆n+ 2κ ∂ t∆n .(27) This still describes a conserved density with finite propagation speed and late-time diffusion, but the crossover is smoothed by the extra gradient term. The diffusive time...

work page 2030

[4] [4]

S. A. Hartnoll and A. P. Mackenzie, Colloquium: Planckian dissipation in metals, Rev. of Mod. Phys.94, 041002 (2022)

work page 2022

[5] [5]

Chowdhury, A

D. Chowdhury, A. Georges, O. Parcollet, and S. Sachdev, Sachdev-ye-kitaev models and be- yond: Window into non-fermi liquids, Rev. of Mod. Phys.94, 035004 (2022)

work page 2022

[6] [6]

Zaanen, Why the temperature is high, Nature430, 512 (2004)

J. Zaanen, Why the temperature is high, Nature430, 512 (2004)

work page 2004

[7] [7]

Kovtun, D

P. Kovtun, D. T. Son, and A. O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett.94, 111601 (2005)

work page 2005

[8] [8]

Maldacena, S

J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, Journal of High Energy Physics2016, 106 (2016)

work page 2016

[9] [9]

Hartman, S

T. Hartman, S. A. Hartnoll, and R. Mahajan, Upper bound on diffusivity, Phys. Rev. Lett. 119, 141601 (2017)

work page 2017

[10] [10]

L. V. Delacretaz, A bound on thermalization from diffusive fluctuations, Nature Physics21, 669 (2025)

work page 2025

[11] [11]

M. Qi, A. Milekhin, and L. Delacr´ etaz, Planckian bound on the local equilibration time, arXiv preprint arXiv:2602.17638 (2026)

work page arXiv 2026

[12] [12]

Planckian bound on quantum dynamical entropy

X. Cao, Planckian bound on quantum dynamical entropy, arXiv preprint arXiv:2507.20914 (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025

[13] [13]

Tsuji, T

N. Tsuji, T. Shitara, and M. Ueda, Bound on the exponential growth rate of out-of-time- ordered correlators, Physical Review E98(2018)

work page 2018

[14] [14]

Pappalardi, L

S. Pappalardi, L. Foini, and J. Kurchan, Quantum bounds and fluctuation-dissipation rela- tions, SciPost Physics12, 130 (2022). 17

work page 2022

[15] [15]

E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Commu- nications in mathematical physics28, 251 (1972)

work page 1972

[16] [16]

Calabrese and J

P. Calabrese and J. Cardy, Time dependence of correlation functions following a quantum quench, Phys. Rev. Lett.96, 136801 (2006)

work page 2006

[17] [17]

A. M. L¨ auchli and C. Kollath, Spreading of correlations and entanglement after a quench in the one-dimensional bose–hubbard model, Journal of Statistical Mechanics: Theory and Experiment2008, P05018 (2008)

work page 2008

[18] [18]

S. R. Manmana, S. Wessel, R. M. Noack, and A. Muramatsu, Time evolution of correlations in strongly interacting fermions after a quantum quench, Phys. Rev. B79, 155104 (2009)

work page 2009

[19] [19]

Kim and D

H. Kim and D. A. Huse, Ballistic spreading of entanglement in a diffusive nonintegrable system, Phys. Rev. Lett.111, 127205 (2013)

work page 2013

[20] [20]

Cheneau, P

M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauß, T. Fukuhara, C. Gross, I. Bloch, C. Kollath, and S. Kuhr, Light-cone-like spreading of correlations in a quantum many-body system, Nature481, 484 (2012)

work page 2012

[21] [21]

Langen, R

T. Langen, R. Geiger, M. Kuhnert, B. Rauer, and J. Schmiedmayer, Local emergence of thermal correlations in an isolated quantum many-body system, Nature Physics9, 640 (2013)

work page 2013

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The relationS Ψ =D Ω ∗S C guarantees the FDT condition Ψ =D Ω ∗C

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Appendix A: The quantum FDT in the time domain In this appendix we collect some results obtained using or deriving different FDT relations in the time domain

The code will be made available at publication. Appendix A: The quantum FDT in the time domain In this appendix we collect some results obtained using or deriving different FDT relations in the time domain. We split the discussion in two parts. The first presents the direct statement of the FDT in terms of a differential operator acting onF, which allows ...

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Assuming F(x, t)≃e −λ tΦ x/(vt) ,(A6) Eq

FromFtoCandR: the differential operator form We define SAB(t) = 1 Z Tr e−βH A(t)B =C AB(t) +ℏR ′′ AB(t),(A1) withC AB(t) andR ′′ AB(t) related to standard fluctuations and response as CAB(t) = 1 2 1 Z Tr e−βH {A(t), B} ,(A2a) RAB(t) = 2iθ(t)R′′ AB(t) = i ℏ θ(t) 1 Z Tr e−βH [A(t), B] .(A2b) The quantum FDT in frequency reads CAB(ω) = cosh βℏω 2 FAB(ω) (A3)...

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[28], Mauger–Pottier derived iℏR ′′ BA(t) = 1 π CBA ∗pv Ω sinh(Ωt) ,(A8) CBA(t) =− iℏ π R′′ BA ∗pv Ω coth(Ωt) ,(A9) where∗denotes convolution

Telescoping and dyadic sum of blurred kernels In Ref. [28], Mauger–Pottier derived iℏR ′′ BA(t) = 1 π CBA ∗pv Ω sinh(Ωt) ,(A8) CBA(t) =− iℏ π R′′ BA ∗pv Ω coth(Ωt) ,(A9) where∗denotes convolution. We wish to relateTΨ AB andC AB, which classically coincide at equilibrium. The FDT relation in frequency reads bΨAB(ω) = 2 ℏω tanh βℏω 2 bCAB(ω),(A10) bCAB(ω) =...

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