# Linearized regime of the generalized hydrodynamics with diffusion

### Submission summary

 As Contributors: Milosz Panfil Arxiv Link: https://arxiv.org/abs/1905.06257v3 (pdf) Date submitted: 2019-09-13 02:00 Submitted by: Panfil, Milosz Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Quantum Physics Approach: Theoretical

### Abstract

We consider the generalized hydrodynamics including the recently introduced diffusion term for an initially inhomogeneous state in the Lieb-Liniger model. We construct a general solution to the linearized hydrodynamics equation in terms of the eigenstates of the evolution operator and study two prototypical classes of initial states: delocalized and localized spatially. We exhibit some general features of the resulting dynamics, among them, we highlight the difference between the ballistic and diffusive evolution. The first one governs a spatial scrambling, the second, a scrambling of the quasi-particles content. We also go one step beyond the linear regime and discuss the evolution of the zero momentum mode that does not evolve in the linear regime.

### Ontology / Topics

See full Ontology or Topics database.

###### Current status:
Has been resubmitted

Dear Editor,

Please find the new version of our manuscript for resubmission.
There is a number of modifications, additions and improvements that comply with the referees' reports.

Yours sincerely,
the authors

### List of changes

The major changes are:

1) improved the general introduction to the GHD and Lieb-Liniger model.
2) changed the inset of Fig. 3, left panel. Instead of plotting the maximal eigenvalue we plot now the trace.
3) added a new figure, Fig. 4, left panel, showing the scaling of eigenvalues of $D_k$.
4) changed Fig. 4, right panel. Instead of ambiguous "Eigenstate index" we label the eigenstates by the imaginary part of the eigenvalues
5) added a discussion on entropy production and its relation to the equilibration in the rapidities space. There is also a new figure, Fig. 8.

We have also improved presentation in a number of places, including the ones suggested by the referees.

### Submission & Refereeing History

Resubmission 1905.06257v4 on 31 October 2019

Resubmission 1905.06257v3 on 13 September 2019
Submission 1905.06257v2 on 10 June 2019

## Reports on this Submission

### Anonymous Report 2 on 2019-10-15 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1905.06257v3, delivered 2019-10-15, doi: 10.21468/SciPost.Report.1227

### Report

I thank the authors for the changes they have made.

The introduction is now clearer, as is most of the discussion.

I now understand what they mean by $\omega(k)$, which has been clarified with figure 4. However, some equations still don’t make sense. The problem is that the authors take the intuition from the case where the k-dependent operators all commute with each other, with a common set of eigenfunctions a different eigenvalues - in this case omega(k) makes sense, as a given eigenfunction can be identified for all operators, with a k-dependent eigenvalue. Here this is not the case. Each operator has its own set of eigenfunctions, and set of eigenvalues. So there is not a unique a priori” way of getting a *function of k* $\omega(k)$ for the eigenvalues - a priori k-independent way of identifying a given eigenvalue. But, from figure 4, what they have done is simply counting the eigenvalue from that with smallest imaginary part to that with largest. This is fine, and indeed gives $\omega(k)$ and one can analyse the scaling.

However, equations (22) and (24) still don’t make sense without further explanations. $\omega$ appears as an integration variable - over the reals? - yet it appears in the exponential as a function of k.

Please adjust equations (22) and (24) or add explanations around, and also clarify from the beginning, from eq (21), what you mean by $\omega(k)$ (e.g. its not “the eigenvalue of $\frak _k$”, because there isn’t a unique eigenvalue, there is rather a continuum).

Once these small points are assessed, the paper can be published.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Milosz Panfil  on 2019-11-04  [id 638]

(in reply to Report 2 on 2019-10-15)
Category:
correction

Dear Referee,

Thank you again for the constructive comments.

We have clarified the presentation in the two places that you have mentioned.

### Report

The authors implemented all the points I raised and I believe the paper can now be published as it stands.

• validity: high
• significance: high
• originality: good
• clarity: high
• formatting: good
• grammar: good