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arxiv: 2605.05604 · v1 · submitted 2026-05-07 · 🪐 quant-ph · cond-mat.stat-mech

Temporal Coarse-Graining as the Origin of Macroscopic Friction in Quantum Spin Chains via Data-Driven Liouvillian Extraction

Seiki Saito This is my paper

Pith reviewed 2026-05-08 11:51 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords quantum spin chainshydrodynamicscoarse-grainingfrictionirreversibilitydata-driven methodsXXZ model
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The pith

Finite temporal coarse-graining turns reversible quantum dynamics into macroscopic friction and viscosity.

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

The paper shows that in an isolated quantum spin chain, the dynamics remain fully reversible when tracked at the finest time resolution. Only when the observer averages over a finite time interval does positive friction and viscosity appear in the extracted hydrodynamic equations. This establishes that macroscopic irreversibility is not built into the microscopic laws but depends on the temporal scale at which the system is observed. The mechanical wave speed stays fixed from the unitary evolution, while dissipation coefficients require the coarse-graining step to become positive. A data-driven method extracts these coefficients directly from simulation data without assuming a form for the friction.

Core claim

By applying generalized extended dynamic mode decomposition to data from the chaotic XXZ spin chain and projecting onto observables that include spin currents, the authors extract the Navier-Stokes coefficients as functions of the coarse-graining timescale. The elasticity coefficient c squared remains positive and independent of coarse-graining, reflecting the underlying unitary evolution. In contrast, the friction gamma and viscosity nu oscillate around zero when the coarse-graining interval approaches zero, but settle to strictly positive values once the interval exceeds a characteristic crossover time. This demonstrates that genuine macroscopic transport coefficients with dissipation are

What carries the argument

The gEDMD procedure combined with Mori-Zwanzig projection applied to an expanded dictionary of spin densities and currents, used to extract hydrodynamic coefficients from finite-time data at varying coarse-graining scales.

If this is right

  • The extracted elasticity is strictly determined by the exact unitary dynamics and does not depend on the observation timescale.
  • Friction and viscosity remain consistent with zero net dissipation in the limit of infinitesimal coarse-graining intervals.
  • A distinct crossover timescale exists beyond which the averaged dynamics produce positive dissipative coefficients.
  • Macroscopic friction is therefore an emergent property controlled by the observer's temporal resolution rather than an intrinsic feature of the microscopic Hamiltonian.

Where Pith is reading between the lines

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

  • The same coarse-graining dependence may appear in other quantum transport problems such as energy or charge flow in isolated systems.
  • Experimental protocols in quantum simulators could test this by deliberately varying the interval between measurements to observe the onset of effective friction.
  • Classical hydrodynamics might be re-derived as the coarse-grained limit of reversible microscopic dynamics in a similar data-driven way.

Load-bearing premise

The assumption that the observable dictionary and gEDMD fitting procedure applied to finite-size simulation data accurately recover the true infinite-system hydrodynamic coefficients without projection artifacts or finite-size effects.

What would settle it

Numerical simulations of the XXZ chain with progressively smaller coarse-graining intervals should show the extracted friction coefficient oscillating around zero with amplitude that does not vanish, or alternatively converging to zero in the exact-derivative limit.

Figures

Figures reproduced from arXiv: 2605.05604 by Seiki Saito.

Figure 1
Figure 1. Figure 1: FIG. 1 view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 view at source ↗
Figure 5
Figure 5. Figure 5: compares the complex plane spectra of the com￾plete basis of all 65, 535 observables (representing the en￾tire system) against the restricted target dictionary (Dict FIG. 5. Genesis of decoherence in an 8-qubit quench (tq = 10.0). The left column shows the spectra obtained us￾ing the complete basis of all 65, 535 observables, while the right column shows the spectra extracted using the restricted target di… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 view at source ↗
read the original abstract

Understanding the emergence of macroscopic irreversible hydrodynamics from the reversible unitary dynamics of isolated quantum many-body systems remains a fundamental challenge. Conventional approaches often force spin density dynamics into purely diffusive models, obscuring the microscopic interplay of pressure, spin current, and local friction. Furthermore, reconciling true irreversibility with strictly unitary evolution raises profound questions about the role of the observer's temporal resolution. In this paper, we introduce a fully data-driven framework based on generalized Extended Dynamic Mode Decomposition (gEDMD) integrated with the Mori-Zwanzig projection. By expanding the observable dictionary to explicitly include spin currents, we directly extract the Navier-Stokes hydrodynamic coefficients from a chaotic XXZ spin chain across varying temporal coarse-graining scales. Our unconstrained extraction reveals a profound physical dichotomy: the mechanical elasticity ($c^2$) is intrinsically derived from the exact unitary dynamics, preserving strict microscopic reversibility. In stark contrast, the macroscopic friction ($\gamma$) and kinematic viscosity ($\nu$) exhibit zero net dissipation, oscillating rapidly around zero in the exact-derivative limit. We demonstrate that genuine macroscopic transport cannot be established without finite temporal coarse-graining. By introducing a finite observation timescale ($\Delta t_{\rm cg} > 0$), the system passes through a distinct crossover timescale where these reversible fluctuations average out, establishing an intermediate functional regime that yields strictly positive friction and viscosity. Our results clearly demonstrate that macroscopic friction in isolated quantum systems is not an absolute property, but fundamentally an emergent phenomenon dictated by the temporal resolution of the observer.

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

2 major / 2 minor

Summary. The paper introduces a data-driven framework combining generalized Extended Dynamic Mode Decomposition (gEDMD) with Mori-Zwanzig projection to extract Navier-Stokes hydrodynamic coefficients (elasticity c², friction γ, and kinematic viscosity ν) directly from unitary trajectories of a chaotic XXZ spin chain. It claims that c² emerges intrinsically from the exact unitary dynamics while γ and ν are strictly zero (oscillating around zero) in the exact-derivative limit, but become positive after finite temporal coarse-graining Δt_cg > 0, establishing that macroscopic friction is an emergent effect of the observer's temporal resolution rather than an intrinsic property of the isolated quantum system.

Significance. If the extraction is robust, the work would be significant for offering a concrete, unconstrained demonstration that irreversibility in quantum transport can arise purely from finite observation timescales without ad-hoc dissipation terms. The expanded observable dictionary and data-driven Liouvillian extraction provide a reproducible route to hydrodynamic coefficients that could be applied more broadly to other many-body systems, strengthening links between unitary quantum dynamics and classical hydrodynamics.

major comments (2)
  1. [Results on hydrodynamic coefficients and crossover] The central claim that γ and ν are strictly zero in the exact-derivative limit but positive for finite Δt_cg rests on the gEDMD procedure faithfully recovering the coefficients without projection artifacts. In finite-N XXZ chains, truncation of memory kernels by the finite observable dictionary and boundary/recurrence effects can mimic or suppress dissipation; the manuscript provides no explicit convergence tests with system size N or dictionary size to rule this out (see results on the crossover timescale).
  2. [Abstract and extraction procedure] No quantitative validation (error bars, multiple independent trajectories, or comparison to known analytic limits of the XXZ model) is reported for the extracted positive γ and ν values at finite Δt_cg. This is load-bearing for the dichotomy between reversible elasticity and dissipative transport, as the abstract notes an unconstrained extraction but the skeptic note highlights the absence of such controls.
minor comments (2)
  1. [Methods] Notation for the coarse-graining timescale Δt_cg and the precise definition of the 'exact-derivative limit' should be clarified with an equation reference to avoid ambiguity in how the Liouvillian is constructed.
  2. [Introduction] The manuscript would benefit from additional references to prior applications of gEDMD and Mori-Zwanzig in quantum spin systems for context on the observable dictionary choice.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our data-driven extraction of hydrodynamic coefficients from unitary XXZ dynamics. We address the two major comments point by point below, clarifying the robustness of our gEDMD+Mori-Zwanzig procedure and committing to added validation where it strengthens the central claim that friction emerges only under finite temporal coarse-graining.

read point-by-point responses

  1. Referee: [Results on hydrodynamic coefficients and crossover] The central claim that γ and ν are strictly zero in the exact-derivative limit but positive for finite Δt_cg rests on the gEDMD procedure faithfully recovering the coefficients without projection artifacts. In finite-N XXZ chains, truncation of memory kernels by the finite observable dictionary and boundary/recurrence effects can mimic or suppress dissipation; the manuscript provides no explicit convergence tests with system size N or dictionary size to rule this out (see results on the crossover timescale).

    Authors: We agree that explicit checks against finite-N and dictionary truncation artifacts are necessary to confirm that the observed crossover is physical rather than numerical. The manuscript already employs the full observable dictionary of spin densities plus currents and reports consistent zero-to-positive transition across the studied chain lengths; however, we did not include systematic scans of N and dictionary size in the main figures. In the revised manuscript we will add a dedicated convergence subsection (with supplementary plots) showing that both the exact-derivative null result for γ,ν and the positive values after Δt_cg remain stable when N is increased and when the dictionary is enlarged, thereby ruling out the suggested projection artifacts. revision: yes

  2. Referee: [Abstract and extraction procedure] No quantitative validation (error bars, multiple independent trajectories, or comparison to known analytic limits of the XXZ model) is reported for the extracted positive γ and ν values at finite Δt_cg. This is load-bearing for the dichotomy between reversible elasticity and dissipative transport, as the abstract notes an unconstrained extraction but the skeptic note highlights the absence of such controls.

    Authors: We acknowledge that quantitative error control and cross-checks are essential for the claimed dichotomy. The extraction is performed on ensembles of unitary trajectories, and the elasticity coefficient c² is already compared internally to the microscopic spin-wave velocity implied by the XXZ Hamiltonian. For the dissipative coefficients at finite coarse-graining, direct analytic benchmarks are unavailable, but we will augment the revised manuscript with (i) explicit error bars obtained from the ensemble variance across independent trajectories and (ii) a new paragraph comparing the extracted c² to the known ballistic transport scale of the model. These additions will be placed in the results section on the crossover timescale. revision: yes

Circularity Check

0 steps flagged

No circularity: data-driven gEDMD extraction demonstrates emergence of friction from finite coarse-graining

full rationale

The paper applies generalized Extended Dynamic Mode Decomposition (gEDMD) with an expanded observable dictionary (including spin currents) to finite-time unitary simulation data of the XXZ chain, then extracts Navier-Stokes coefficients as a function of the explicit coarse-graining timescale Δt_cg. The reported zero net dissipation in the exact-derivative limit follows directly from the unitary input trajectories, while the strictly positive γ and ν at finite Δt_cg are outputs of the projection procedure rather than inputs. No self-citations, uniqueness theorems, or ansatzes are load-bearing; the dictionary choice and Mori-Zwanzig integration are stated explicitly and the extraction is described as unconstrained. The derivation chain therefore remains self-contained against the simulation data and does not reduce to its own assumptions by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the chosen observable dictionary plus gEDMD projection accurately isolates hydrodynamic modes from microscopic unitary evolution. No new particles or forces are postulated; the only free parameter is the coarse-graining window itself, which is varied rather than fitted to a target value.

free parameters (1)
  • coarse-graining timescale Δt_cg
    Varied as an external parameter to reveal the crossover; not fitted to match a specific dissipation value.
axioms (2)
  • standard math The XXZ spin chain dynamics remain strictly unitary for all simulated times.
    Invoked when stating that friction must average to zero in the exact-derivative limit.
  • domain assumption The gEDMD dictionary expansion with spin currents is sufficient to capture Navier-Stokes level hydrodynamics.
    Required for the extraction procedure to be interpreted as yielding macroscopic coefficients.

pith-pipeline@v0.9.0 · 5572 in / 1620 out tokens · 37828 ms · 2026-05-08T11:51:45.311023+00:00 · methodology

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    Our data-driven framework thus uniquely visualizes the exact physical crossover where ab- solute reversible mechanics gives way to emergent macro- scopic hydrodynamics

    and artificial over-damping due to the sinc-filter effect at larger timescales. Our data-driven framework thus uniquely visualizes the exact physical crossover where ab- solute reversible mechanics gives way to emergent macro- scopic hydrodynamics. In conclusion, we have introduced a comprehen- sive framework combining Exact-Derivative and finite- differe...

    work page

  2. [2]

    Colloquium: Nonequi- librium dynamics of closed interacting quantum systems

    Anatoli Polkovnikov, Krishnendu Sengupta, Alessandro Silva, and Mukund Vengalattore. Colloquium: Nonequi- librium dynamics of closed interacting quantum systems. Rev. Mod. Phys., 83:863–883, 2011

    work page 2011

  3. [3]

    Equilibration, ther- malisation, and the emergence of statistical mechanics in closed quantum systems.Reports on Progress in Physics, 79(5):056001, 2016

    Christian Gogolin and Jens Eisert. Equilibration, ther- malisation, and the emergence of statistical mechanics in closed quantum systems.Reports on Progress in Physics, 79(5):056001, 2016

    work page 2016

  4. [4]

    J. M. Deutsch. Quantum statistical mechanics in a closed system.Phys. Rev. A, 43:2046–2049, 1991

    work page 2046

  5. [5]

    Chaos and quantum thermalization

    Mark Srednicki. Chaos and quantum thermalization. Phys. Rev. E, 50:888–901, 1994

    work page 1994

  6. [6]

    Ther- malization and its mechanism for generic isolated quan- tum systems.Nature, 452(7189):854–858, 2008

    Marcos Rigol, Vanja Dunjko, and Maxim Olshanii. Ther- malization and its mechanism for generic isolated quan- tum systems.Nature, 452(7189):854–858, 2008

    work page 2008

  7. [7]

    Subroto Mukerjee, Vadim Oganesyan, and David A. Huse. Statistical theory of transport by strongly inter- acting lattice fermions.Phys. Rev. B, 73:035113, 2006

    work page 2006

  8. [8]

    Finite-temperature transport in one- dimensional quantum lattice models.Rev

    Bruno Bertini, Fabian Heidrich-Meisner, Christoph Karrasch, Tomaˇ z Prosen, Robin Steinigeweg, and Marko ˇZnidariˇ c. Finite-temperature transport in one- dimensional quantum lattice models.Rev. Mod. Phys., 93:025003, 2021

    work page 2021

  9. [9]

    Spin diffusion from an interacting quantum system.Na- ture Communications, 8(1):16117, 2017

    Marko Ljubotina, Marko ˇZnidariˇ c, and Tomaˇ z Prosen. Spin diffusion from an interacting quantum system.Na- ture Communications, 8(1):16117, 2017

    work page 2017

  10. [10]

    Garrahan

    Emanuele Levi, Markus Heyl, Igor Lesanovsky, and Juan P. Garrahan. Robustness of many-body localiza- tion in the presence of dissipation.Phys. Rev. Lett., 116:237203, 2016

    work page 2016

  11. [11]

    Williams, Ioannis G

    Matthew O. Williams, Ioannis G. Kevrekidis, and Clarence W. Rowley. A data–driven approximation of the Koopman operator: Extending dynamic mode decompo- sition.Journal of Nonlinear Science, 25(6):1307–1346, 2015

    work page 2015

  12. [12]

    Data- driven model reduction and transfer operator approxi- mation.Journal of Nonlinear Science, 28(3):985–1010, 2018

    Stefan Klus, Feliks N¨ uske, P´ eter Koltai, Hao Wu, Ioan- nis Kevrekidis, Christof Sch¨ utte, and Frank No´ e. Data- driven model reduction and transfer operator approxi- mation.Journal of Nonlinear Science, 28(3):985–1010, 2018

    work page 2018

  13. [13]

    Amari Muhamad, and Frank No´ e

    Stefan Klus, Ingmar Schuster, S. Amari Muhamad, and Frank No´ e. Data-driven approximation of the Koopman generator: Model reduction, system identification, and control.Physica D: Nonlinear Phenomena, 406:132416, 2020

    work page 2020

  14. [14]

    Transport, collective motion, and brown- ian motion.Progress of Theoretical Physics, 33(3):423– 455, 1965

    Hazime Mori. Transport, collective motion, and brown- ian motion.Progress of Theoretical Physics, 33(3):423– 455, 1965

    work page 1965

  15. [15]

    Nonlinear generalized langevin equa- tions.Journal of Statistical Physics, 9(3):215–220, 1973

    Robert Zwanzig. Nonlinear generalized langevin equa- tions.Journal of Statistical Physics, 9(3):215–220, 1973

    work page 1973

  16. [16]

    Al-Mohy and Nicholas J

    Awad H. Al-Mohy and Nicholas J. Higham. Computing the action of the matrix exponential, with an application to exponential integrators.SIAM Journal on Scientific Computing, 33(2):488–511, 2011

    work page 2011