# Perfect Fluids

### Submission summary

 Authors (as Contributors): Jelle Hartong · Niels Obers · Stefan Vandoren · Jan de Boer
Submission information
Date submitted: 2018-01-16 01:00
Submitted by: Vandoren, Stefan
Submitted to: SciPost Physics
Ontological classification
Specialties:
• Condensed Matter Physics - Theory
• Fluid Dynamics
Approach: Theoretical

### Abstract

We develop a new theory of perfect fluids with translation and rotation symmetry, which is also applicable in the absence of any type of boost symmetry. It involves introducing a new fluid variable, the kinetic mass density, which is needed to define the most general energy-momentum tensor for perfect fluids. Our theory leads to corrections to the Euler equations for perfect fluids that might be observable in hydrodynamic fluid experiments. We also derive new expressions for the speed of sound in perfect fluids. Our theory reduces to the known perfect fluid models when boost symmetry is present. It can also be adapted to (non-relativistic) scale invariant fluids with critical exponent $z$. We show that perfect fluids cannot have Schr\"odinger symmetry unless $z=2$. For generic values of $z$ there can be fluids with Lifshitz symmetry, and as a concrete example, we work out in detail the thermodynamics and fluid description of an ideal gas of Lifshitz particles and compute the speed of sound for the classical and quantum Lifshitz gasses.

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

### Submission & Refereeing History

Resubmission 1710.04708v3 on 15 May 2018

Submission 1710.04708v2 on 16 January 2018

## Reports on this Submission

### Anonymous Report 2 on 2018-2-22 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1710.04708v2, delivered 2018-02-22, doi: 10.21468/SciPost.Report.360

### Strengths

I think this paper has the potential to provide a modern treatment of hydrodynamics without Lorentz/Galilean invariance -- see report.

### Weaknesses

I think the presentation at the moment is rather poor and will not get across the main points -- see other referee's report, and my report.

### Report

I agree, more or less, with most of the comments of the prior referee, so I will not repeat them. My one point of disagreement is that I think 1612.07324 does not fully solve the hydrodynamic problem of interest here, and so there is a role for a paper of this form to exist -- and to have genuinely new physics in it.

However, the manuscript at present falls very far short of what it could be. I think some of what follows is subjective, so I would be willing to consider the response of the authors. But, my view is this: the way that fluids are introduced in this paper is essentially the way it would have been done in 1960 (if not earlier). It is a shame that the modern way of introducing fluid dynamics is relegated to a short section 2.8. Rather than painfully detailing the differences between Lorentz/Galilean/Carollian/etc. invariances...the authors really should introduce the geometric perspective much sooner and simply explain how the stronger boost symmetries cause the partition function to be more constrained than it would have otherwise been. I think an expanded geometric discussion section 2.8 should form the heart of section 2 -- almost all of the rest of section 2, in particular 2.4 and possibly 2.5-7, should be relegated to appendices. The current presentation makes the paper bulky and buries the interesting new perspectives that this paper gives.

In my view, most of Sections 3 and 4 are not particularly new, or interesting -- Section 3 for example could be put as an appendix. What the authors should do in section 4, in my view, is to remove a lot of the unnecessary clutter about the Galilean/Lorentzian limits, and to focus on a simple point. The non-trivial thing about the hydrodynamics in the absence of boosts -- which was not carefully understood/explained in 1612.07324 -- is the way that the pressure depends on the velocity at the nonlinear level. Besides the elegant geometric argument of section 2.8, the punchline of this paper should be an explanation/demonstration of whether or not the pressure can obtain interesting and subtle v^2 dependence. The authors have certainly put in some work along these lines but, as in section 2, there is just so much calculation about the z=1,2 limits that the interesting new physics feels completely buried. I also feel that the velocity dependence of pressure for general z is not properly unpacked at present. For example, in the limit of low velocities, relativistic and Galilean hydrodynamics can look "similar" (see, e.g. 0809.4512). Can one make a similar statement for general z? What is the velocity dependence of pressure at large z? These are the obvious questions one would ask that go beyond the textbook fluid theory, and this paper needs to confront them directly. This perspective is what would make this paper stand out from what has been understood for a long time.

### Requested changes

See report.

• validity: good
• significance: high
• originality: high
• clarity: low
• formatting: reasonable
• grammar: excellent

### Anonymous Report 1 on 2018-2-10 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1710.04708v2, delivered 2018-02-10, doi: 10.21468/SciPost.Report.341

### Strengths

See report below.

### Weaknesses

See report below.

### Report

I have written several papers in the past few years using hydrodynamics without any kind of boost symmetry, and I view it as a more or less trivial generalization of "standard" hydrodynamics -- just with less structure and hence with more independent transport coefficients and thermodynamic susceptibilities.

The claim in the abstract of this paper to develop a "new theory of perfect fluids ... in the absence of any kind of boost symmetry" therefore made me somewhat skeptical. Unfortunately this skepticism did not diminish as I starting reading the paper.

Most of the conceptual structure developed at the start of this paper can be found in section 5.4.5 (and other sections) of the review https://arxiv.org/abs/1612.07324. In particular, the "new fluid variable" defined in equation 2.8 of the paper under review is the same as that appearing in equation 459 of the aforementioned review. This quantity is referred to as M or as chi_PP in many previous works. The inclusion of v.dp terms in discussion of thermodynamic variations can also be found there and in other papers (often rather briefly, the question is to what extent a more elaborate exposition is necessary). The sound speed -- the second "main result" of the paper under review -- will follow immediately from the hydrodynamical constitutive relations, conservations laws, and thermodynamic relations. The consequences of scaling ward identities on thermodynamic variables is also rather simple and previously discussed in several places (including the review above).

Part of the problem of the paper is that it seeks to establish a contrast (in e.g. the first few pages) with the way hydrodynamics is usually set up, supposedly built around Galilean or Lorentzian boosts. However, this is not really the right way to set up hydrodynamics even in those cases -- the basic symmetries are translations that guarantee the existence of the hydrodynamical velocity field. They authors may find useful the development of hydrodynamics in e.g. the textbook by Chaikin and Lubensky. While that book does restrict to Galilean-invariant systems for the most part, the boost symmetry is not instrumental in the logic, but just sets certain transport coefficients to zero.

The above said, there are fairly widespread misconceptions in both the high energy and condensed matter communities regarding the necessity of (respectively) Lorentzian or Galilean symmetries for hydrodynamics, and this paper is somewhat symptomatic of these misconceptions, even while trying to confront them. It is also true that there is not much work done on showing how e.g. Lorentzian hydrodynamics emerges from a more general structure, especially at the nonlinear level. These are questions that the paper under review addresses to some extent.

Therefore, this could be a useful paper, at the very least sociologically. I think, though, in order for it to be useful it should be fairly majorly re-written. The are two main aspects of this. Firstly, the overall framing should be more modest and contextualized by previous work, explaining to and reminding the reader that hydrodynamics just requires enough symmetries to guarantee the existence of the hydrodynamic modes and not more (as is well known by at least some people, and in some textbooks). Secondly, many of the derivations involve rather elementary thermodynamic etc. manipulations that are spelt out in some detail and applied to several different cases. Much of this could be relegated to appendices, so that the text could bring out the more conceptual way in which certain specific systems, with additional symmetries beyond the bare ones, fit into the more general structure. (Tangentially, there is also a large literature on which symmetries create new hydrodynamic modes and which ones don't -- this shows up especially in the context of symmetry breaking, e.g. why when you spontaneously break translations and rotations simultaneously you don't get goldstones for rotations, only for translations. This brings out forcefully the difference between the role of e.g. translational and boost symmetries).

Once this is done, this will not be an especially innovative paper but it will be one that helps to clarify the nature of hydrodynamics and may well help to reduce confusion in several fields.

### Requested changes

See report.

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

### Author:  Stefan Vandoren  on 2018-03-12  [id 228]

(in reply to Report 1 on 2018-02-10)

We have been processing the referee reports and would like to make the following comments.

1. As for Referee 1, we would like to stress that because of the absence of boost symmetry, different inertial observers will measure different values for the hydrodynamical observables and quantities. That is one of the reasons why one should compute all the velocity dependent terms in the partition function. Referee 2 clearly points this out, in reply to Referee 1. The relevance of this is also demonstrated in the calculation of the speed of sound for Lifshitz gases. (Applying the wrong formula for Lifshitz sound speeds has appeared in many other papers, leading to the wrong formula $c_s^2=z/d$.)
2. To Referee 1, it is true that the concept of ‘kinetic mass density’ has appeared before, such as in the review 1612.07324. We are happy to include that in our references. But, it is not stressed there that this quantity depends on the frame of the observer. In fact, this quantity only appears in the literature at zero background velocity. Furthermore, as a general comment, that same review states explicitly (see page 97): “…we do not have a systematic understanding of how this arises. To the best of our knowledge, there has not been a systematic development of hydrodynamics for theories that are neither Galilei nor Lorentz invariant.” This is precisely what we started in our paper. If Referee 1 wants us to change “new theory” into “systematic development”, that is fine with us.
3. Writing papers for sociological purposes may sometimes be useful, but is not our primary goal. There is enough new material in the paper, as stated also by Referee 2. It is correct that our paper could have been written decades ago, but it never was.
4. Referee 1 ticks the boxes “High Validity”, “Good Clarity” and “Good Formatting”, yet he/she asks us to majorly rewrite the paper. In our opinion, the paper is well written and accessible to a broad audience, and the authors are experienced enough to give a clear exposition. Moreover, Referee 2 also wants us to rewrite stuff, but wants us to emphasize different things, in tension with what Referee 1 wants, as far as we understand. We believe authors should have the freedom about how they want to present their results and what should be emphasized, as long as it is clear and meeting the scientific standards of the journal. Perhaps the calculation about the speed of sound in a non-zero velocity dependent background could be put to an appendix, we did discuss that before among the authors.
5. As for Referee 2, in our opinion Section 2.5, 2.6 and 2.7 are as important as 2.8. One can argue about 2.4 but it is all subjective and we chose it to do this way.
6. Referee 2 also states that Section 4 contains nothing new or interesting, so please point out for us e.g. where the correct sound speed for Lifshitz gases appears. It is discussed in many places in the literature, and they are all wrong.

Concretely, our proposal is to make the following changes:

1. Rephrase ‘new theory’ into ‘systematic development’ throughout the paper. We keep emphasizing though that computing velocity dependence is important and is one of the new results of the paper. For instance, besides the speed of sound, the no-go theorem about no Schrodinger perfect fluids for z\neq 2 is a nice application of studying velocity dependence, and also a new result. Furthermore, we add some references to the review mentioned above.
2. We will move the calculation of the speed of sound in a non-zero velocity dependent background in an appendix, and some details of Section 4 as well if insisted.
3. We will emphasize a bit more the velocity dependence of the pressure, following the request of Referee 2. It was actually needed in Section 4, so it is easy to make that a bit more visible. All the velocity dependence in the grand partition function is written in section 4, so one can elaborate a bit more on the hydrodynamic quantities here.

Hereby, we are trying to look for a compromise. We do not see much added value by a major rewriting of the draft, as we then rather withdraw. Of course, if the editors/referees agree, we will prepare the file and submit it again for approval.