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Chapman-Enskog theory and crossover between diffusion and superdiffusion for nearly integrable quantum gases
by Maciej Łebek, Miłosz Panfil
Submission summary
| Authors (as registered SciPost users): | Milosz Panfil · Maciej Łebek |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2410.23209v2 (pdf) |
| Date submitted: | April 10, 2025, 10:32 a.m. |
| Submitted by: | Łebek, Maciej |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Integrable systems feature an infinite number of conserved charges and on hydrodynamic scales are described by generalised hydrodynamics (GHD). This description breaks down when the integrability is weakly broken and sufficiently large space-time-scales are probed. The emergent hydrodynamics depends then on the charges conserved by the perturbation. We focus on nearly-integrable Galilean-invariant systems with conserved particle number, momentum and energy. Basing on the Boltzmann approach to integrability-breaking we describe dynamics of the system with GHD equation supplemented with a collision term. The limit of large space-time-scales is addressed using Chapman-Enskog expansion adapted to the GHD equation. For length scales larger than ∼λ−2, where λ is integrability-breaking parameter, we recover Navier-Stokes equations and find transport coefficients: viscosity and thermal conductivity. At even larger length scales, this description crosses over to Kardar-Parisi-Zhang universality class, characteristic to generic non-integrable one-dimensional fluids. Employing nonlinear fluctuating hydrodynamics we estimate this crossover length scale as ∼λ−4.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Thank you for taking care of our submission.
We are aware that our manuscript has received criticism from the two Referees. However, we are convinced that with this revised manuscript we were able to fully address the points raised in both reports. Therefore, we would like to ask you to reconsider submitting our article once again to the SciPost Physics.
Importantly, motivated by the comments of both Referees, we have decided to expand our article and include a new section about the emergence of anomalous transport in 1d fluids. This section contains essential new results, which are of major importance for the topic of weak integrability breaking and address the doubts of the Second Referee. Their objection assumes that the transport in 1d is always anomalous despite the counterexamples that we gave in the Conclusions to the first version of this manuscript. Those counterexamples were pointing towards a crossover regime between a normal and anomalous hydrodynamics. In this new section we present a general argument for the existence of the crossover by employing the non-linear fluctuating hydrodynamics. This is the very method that was used to argue for the anomalous hydrodynamics in the first place and shows that there is no contradictions in existence of these two regimes. The relevance of the additional content is reflected in the modified title of the manuscript, which highlights now also the crossover between diffusive and superdiffusive transport.
We strongly believe that our article, due to these improvements, fulfills the criteria of the journal. We hope that the Referees, once they see the recent changes, will agree with us on that point.
Sincerely,
Authors
List of changes
1. We changed the title of our manuscript to "Chapman-Enskog theory and crossover between diffusion and superdiffusion for nearly integrable quantum gases".
2. We added a new section (Sec.5 in the present version) which discusses the crossover between Navier-Stokes diffusive transport and anomalous, Kardar-Parisi-Zhang superdiffusion.
3. We modified the abstract which presents now a short summary of the new section.
4. We modified the introduction and the summary to incorporate the new findings of the second version.
5. We added Figure 1, which presents crossovers between different hydrodynamic regimes of nearly integrable quantum gases.
6. We added Figure 2 presenting numerical solutions to the mode-coupling equations.
7. We added Appendix B with material related to the content of new section.
List of corrections:
1. We introduced abbreviation for "Chapman-Enskog" as "ChE" and "Navier-Stokes" as "NS"
2. We corrected typos in Eqs. (15), (127).
3. We updated the references which in present enumeration are [30], [51] and added references [66], [80], [81], [82], [84], [88].
Current status:
Reports on this Submission
Report
The authors made substantial additions to the paper, which address the concerns I raised before. I find their results regarding the crossover between normal and anomalous hydrodynamic behaviour especially interesting. Therefore, I have no hesitation in recommending their work for publication in Scipost Physics.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report
I commend the authors for their detailed and thoughtful engagement with my comments. I agree with the authors that their new results predicting an explicit crossover timescale from a “conventional” hydrodynamic regime to an anomalous regime are novel and potentially of broader interest than the results reported in the initial draft. In particular, it was not at all clear from the conclusion of the previous draft that a parametrically large crossover region between normal and anomalous hydrodynamics could exist. The authors have now provided a thorough and detailed argument that such a regime does in fact exist. This both a posteriori justifies the Chapman-Enskog approach taken by the authors, and strikes me as interesting, physically insightful and concrete enough to be tested in future work. I am therefore happy to recommend the revised manuscript for publication in Scipost.
I did have a couple of follow-up remarks that the authors might consider:
1. Chapman-Enskog is indeed expected to yield accurate (if uncontrolled) results in weakly coupled 3D systems. But this a very different regime of approximation from 1D and strong coupling, which is why the authors’ predictions are still a hypothesis to be tested. This limitation of Chapman-Enskog is not really addressed in the text.
2. The terminology of “GHD-Boltzmann” equations sounds redundant to me. The viewpoint that the GHD equation should be viewed as a Boltzmann equation without dissipative collision terms was explicitly proposed in Phys. Rev. B 97, 045407 (2018) and Phys. Rev. Lett. 120, 045301 (2018) and is implicit in Ref. 29. The authors also remark that “Although GHD-Boltzmann equation differs from the standard Boltzmann counterpart by the presence of nonlinear effective velocity in the streaming term and by diffusion of quasiparticles” but in fact both of these effects have been standard for decades in the kinetic theory of Fermi liquids, which is introduced e.g. in Landau and Lifshitz Vol. 9, Chapter 1.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)