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arxiv: 2408.15163 · v5 · submitted 2024-08-27 · ❄️ cond-mat.quant-gas · nlin.CD· physics.atom-ph· physics.flu-dyn

Turbulence and far-from-equilibrium equation of state of Bogoliubov waves in Bose-Einstein Condensates

Ying Zhu , Giorgio Krstulovic , Sergey Nazarenko This is my paper

Pith reviewed 2026-05-23 21:24 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas nlin.CDphysics.atom-phphysics.flu-dyn
keywords Bogoliubov wavesBose-Einstein condensateswave turbulenceKolmogorov spectrumequation of stateGross-Pitaevskiiturbulent cascadesfar-from-equilibrium
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The pith

A new Kolmogorov-like spectrum for short Bogoliubov waves explains the far-from-equilibrium equation of state in Bose-Einstein condensates.

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

The authors apply wave turbulence theory to Bogoliubov waves in BECs by deriving stationary spectra from the wave-kinetic equations of the Gross-Pitaevskii model. They obtain a power-law Kolmogorov-like spectrum for short waves and a full analytical form for the long-wave acoustic regime. These spectra are then used to account for the equation of state measured in a recent experiment with driven BECs. This matters because it provides a theoretical framework for understanding energy distributions and thermodynamic properties in non-equilibrium quantum gases.

Core claim

Using the 3D Gross-Pitaevskii model and associated wave-kinetic equations, we derive a new Kolmogorov-like stationary spectrum for short Bogoliubov waves and a complete analytical expression for the spectrum in the long-wave acoustic regime. These predictions explain the BEC equation of state reported in experiments.

What carries the argument

Kolmogorov-like stationary spectrum for short Bogoliubov waves derived from wave-kinetic equations.

If this is right

  • The derived spectrum predicts the energy distribution across wave lengths in turbulent BECs.
  • It provides an explanation for the observed far-from-equilibrium equation of state in driven condensates.
  • New experimental settings are suggested to further test the acoustic regime predictions.

Where Pith is reading between the lines

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

  • This framework could be extended to study turbulence in other systems with similar dispersion relations.
  • Direct observation of the predicted spectra in experiments would strengthen the link between wave turbulence theory and BEC observations.
  • Similar methods might apply to understanding non-equilibrium states in other quantum fluids.

Load-bearing premise

The wave-kinetic equations remain valid and the system reaches a stationary turbulent state under the driving and dissipation conditions of the Dogra et al. experiment.

What would settle it

If measurements in a BEC experiment show a wave spectrum that does not match the predicted Kolmogorov-like form for short waves or the analytical expression for long waves, the central claim would be falsified.

Figures

Figures reproduced from arXiv: 2408.15163 by Giorgio Krstulovic, Sergey Nazarenko, Ying Zhu.

Figure 1
Figure 1. Figure 1: FIG. 1: Numerical results obtained by simulating WKE [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: EoS generated from the experimental data of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

Bogoliubov waves are fundamental excitations of Bose-Einstein Condensates (BECs). They emerge from a perturbed ground state and interact nonlinearly, triggering turbulent cascades. Here, we study turbulent BECs theoretically and numerically using the 3D Gross-Pitaevskii model and its associated wave-kinetic equations. We derive a new Kolmogorov-like stationary spectrum for short Bogoliubov waves and find a complete analytical expression for the spectrum in the long-wave acoustic regime. We then use our predictions to explain the BEC equation of state reported by Dogra et al. (Nature 620, 521, 2023), and to suggest new experimental settings.

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

1 major / 1 minor

Summary. The manuscript derives a new Kolmogorov-like stationary spectrum for short Bogoliubov waves and a complete analytical expression for the spectrum in the long-wave acoustic regime from the 3D Gross-Pitaevskii model and associated wave-kinetic equations. It then applies these predictions to explain the equation of state measured by Dogra et al. (Nature 620, 521, 2023) and proposes new experimental settings.

Significance. If the central derivations hold and the application to experiment is valid, the work supplies analytical predictions for turbulent spectra in BECs that can be tested directly, strengthening the link between weak-turbulence theory and far-from-equilibrium measurements in quantum gases. The provision of complete analytical expressions without free parameters is a clear strength.

major comments (1)
  1. [Application to Dogra et al. experiment] Application section (comparison with Dogra et al.): the claim that the derived short-wave Kolmogorov-like spectrum and long-wave acoustic expression explain the measured BEC equation of state requires that the wave-kinetic equations reach a stationary state under the specific driving and dissipation of the 2023 experiment. No direct numerical test with those forcing/damping parameters is reported to confirm stationarity or spectral shape, leaving the applicability open to question from trap inhomogeneity or stronger nonlinearity.
minor comments (1)
  1. [Abstract] Abstract: states that derivations exist but provides no equation numbers or brief statement of the key assumptions (e.g., weak-turbulence closure), reducing immediate clarity for readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive feedback. We provide a point-by-point response to the major comment.

read point-by-point responses

  1. Referee: [Application to Dogra et al. experiment] Application section (comparison with Dogra et al.): the claim that the derived short-wave Kolmogorov-like spectrum and long-wave acoustic expression explain the measured BEC equation of state requires that the wave-kinetic equations reach a stationary state under the specific driving and dissipation of the 2023 experiment. No direct numerical test with those forcing/damping parameters is reported to confirm stationarity or spectral shape, leaving the applicability open to question from trap inhomogeneity or stronger nonlinearity.

    Authors: We acknowledge that our manuscript does not include direct numerical simulations of the wave-kinetic equations using the precise driving and dissipation parameters from the Dogra et al. (2023) experiment. Our numerical validations are performed in controlled settings to confirm the analytical spectra. The stationary nature of the derived spectra follows directly from solving the wave-kinetic equation in the appropriate asymptotic regimes, and these solutions are expected to be attractors under weak driving and dissipation as long as the weak turbulence assumptions hold. The experimental data from Dogra et al. show power-law behaviors consistent with our predictions, supporting the interpretation of the equation of state. We maintain that trap inhomogeneity is accounted for in the long-wave regime through the acoustic spectrum, and the nonlinearity remains weak enough for the kinetic description to apply, as indicated by the agreement with the parameter-free theory. If the referee deems it necessary, we can revise the manuscript to include a more detailed discussion of these assumptions and the conditions under which stationarity is reached. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivations are independent of target data.

full rationale

The paper derives new stationary spectra analytically from the wave-kinetic equations of the 3D Gross-Pitaevskii model under weak-turbulence assumptions, presenting them as first-principles results rather than fits. The subsequent use of these spectra to interpret the Dogra et al. equation-of-state data is an application of the derived expressions, not a redefinition or statistical forcing of the spectra by the data. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claims to tautologies are present. The derivation chain remains self-contained against the model equations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted. The wave-kinetic closure and stationarity assumption are implicit but not itemized.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Strong and weak wave turbulence regimes in Bose-Einstein condensates

    cond-mat.quant-gas 2024-11 unverdicted novelty 6.0

    Simulations of inverse-cascade turbulence in 3D BECs reveal forcing-dependent regime transitions and yield a new out-of-equilibrium equation of state.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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    = δ(ωk − ωk1 − ωk2) , (5) where V k 12 ≡ V (k, k1, k2) is the three-wave interaction amplitude depending on the particular type of waves. For Bogoliubov waves, they are given by V 1 23 = V0 p k1k2k3W 1 23 , V 0 = 3√cs 4 √ 2 , W 1 23 = 1 2√η1η2η3 + √η1η2η3 6k1k2k3 k3 1 η1 − k3 2 η2 − k3 3 η3 , (6) where ηi ≡ η(ki) = p 1 + (kiξ)2/2 for i = 1 , 2, 3. The Bog...

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    (7) − |W 1 k2|2T 1 k2δ(ω1 k2) − |W 2 k1|2T 2 k1δ(ω2 k1) k2 1k2 2dk1dk2 , where T k 12 = nk1 nk2 − nknk1 − nknk2. In Appendix A, a revised, rigorous derivation of the WKE, its asymptotic regimes, and the parametrization of the resonant mani- fold can be found. In this Letter, we obtain new theoretical predictions for the 3D Bogoliubov WTT in both large- an...

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    Hamiltonian formation and wave interaction amplitude V 1 23 The action for Bogoliubov waves (per unit of mass) is expressed in hydrodynamic variables it is given by S = 1 ρ0L3 Z dtd3x −ρ ˙ϕ − ρ 2(∇ϕ)2 − c2 s 2ρ0 (ρ − ρ0)2 − c2 s ξ2 (∇√ρ)2 , (A1) which corresponds to a compressible, isentropic, irrotational fluid [22] but with an extra quantum pressure ter...

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    A different method to obtain the interaction coefficient: the Bogoliubov transformation A different way to find the interaction coefficient V 1 23 exploits the Bogoliubov transformation. The starting point is the Hamiltonian (per unit of mass) for the GPE in terms of Ψ and ρ. It reads H = 1 L3ρ0 Z c2 s ξ2|∇Ψ|2 + c2 s 2ρ0 (ρ − ρ0)2 d3x . (A8) Consider weak...

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    Angular average of δ(k − k1 − k2) To get the isotropic WKE, we only need to compute the angular average of the Dirac- δ function of wavevectors in Eq. (5), since V 1 23 and frequency are both angle-independent. Writing d k = k2dΩdk, where dΩ = sin θdθdϕ, the angular average of δ(k − k1 − k2) is then defined as ∆ = R δ(k − k1 − k2)dΩ1dΩ2. We define a coord...

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