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As Contributors: | Jan de Boer · Stefan Vandoren · Niels Obers · Jelle Hartong |

Arxiv Link: | https://arxiv.org/abs/1710.04708v3 |

Date accepted: | 2018-06-19 |

Date submitted: | 2018-05-15 |

Submitted by: | Vandoren, Stefan |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | Fluid Dynamics |

We present a systematic treatment 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 fluid variable, the {\em kinetic mass density}, which is needed to define the most general energy-momentum tensor for perfect fluids. Our analysis 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 that reduce to the known perfect fluid models when boost symmetry is present. Our framework 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 gases.

Published
as
SciPost Phys. **5**, 003
(2018)

Changes/Modifications in the Draft:

1) Abstract/Intro: changed the phrasing “develop a new theory” into “present a systematic or unified analysis/treatment/framework”.

2) Added in intro and in section 2 a few sentences mentioning that the notion of kinetic mass density was already discussed in the literature (e.g. review 1612.07324), but at the same time stress that this was at zero velocity (lab frame), whereas we want to know it in all frames. Moreover, we stressed in the intro that this kinetic mass density follows from the pressure at finite velocity. Therefore, it is important to determine velocity dependent terms.

3) Streamlined in the intro better the three main new results obtained in this paper, stressing also the unified picture we give that can be applied to all perfect fluids with and without boost symmetry.

4) Restructured Section 3, and moved previous section 3.2 (boosted fluids) to an appendix.

5) Major editing in Section 4: deleted all the discussion of the special values of z=1 and z=2, such that the exposition is shorter and flows more smoothly. Furthermore, we added some more text about the velocity dependent terms, for instance in the pressure.

6) Added a discussion about the large z-limit, and the velocity dependent terms in that limit, also in Section 4.

Resubmission 1710.04708v3 (15 May 2018)

- Report 3 submitted on 2018-06-14 22:16 by
*Anonymous* - Report 2 submitted on 2018-06-05 13:51 by
*Anonymous* - Report 1 submitted on 2018-06-03 07:31 by
*Anonymous*

Submission 1710.04708v2 (16 January 2018)

See below

See below

The authors have improved the paper, but have not taken to heart the comments of, at this point, three referees. The comments of these referees are all broadly in agreement and relate to the novelty of the ideas presented. That said, novel or not, the technical content of the paper appears to be correct and may be helpful to some readers. Therefore, I think that it is publishable at this point.

Perhaps I can best illustrate the reservation as follows. The authors emphasize that they are treating the velocity as a chemical potential. In fact, I would have said that is the definition of the velocity (as it appears in e.g. their 2.1), and it's certainly not new. Given this fact, it is obvious that in the absence of boost symmetries the pressure will depend on the velocity. Relatedly, while it's conceivable that 2.10 has never been written down in this form (as the authors claim somewhat boldly in footnote 3 -- have they checked every paper in existence?), in any case it's a trivial consequence of the above fact. On this point, and this just happens to be one paper that I know off the top of my head, the authors may find equation 9 of https://arxiv.org/abs/0809.4870 instructive. Chi in that equation is the superfluid velocity squared, which is indeed not associated to a boost symmetry, and so this equation has a rather strong analogy to 2.9 in the paper under review. Obviously the equations are not the same equation, one has to do with superfluids and the other doesn't, but the point is that once thermodynamics can depend on a velocity this kind of relation is a rather immediate thing that drops out.

Perhaps I would mind less if the authors were able to present these points without claiming novelty at every turn. The word "new" appears 16 times in the paper. Some journals have a policy of not allowing explicit use of "new" and "novel" etc., and perhaps that would help here.

The manuscript is well written and for the most part easy to digest.

The distinction between known results and new results is not made clearly enough.

The manuscript reminds me of an American Journal of Physics style article. I suspect the content is for the most part not new, but fills what appears to be a pedagogical gap in the literature on hydrodynamics in systems without boost symmetry. The authors treat in detail the ideal case, i.e. the case without dissipation. (A fuller treatment is promised in refs. [3] and [4].) The ``new'' results surround the dependence of hydrodynamic quantities on the ``kinetic mass density''. The authors compute the speed of sound in their framework, and also provide a ``no go'' theorem for fluids with Schrodinger symmetry. They also discuss in detail a simple example -- an ideal gas of Lifshitz particles.

Regarding the newness of (2.10), I would point out that a similar quantity shows up in the hydrodynamics of superfluids, where v is replaced by the gradient of the phase of the order parameter. I wonder if (2.10) may show up more generally in the hydrodynamics of two-component fluids.

1) On page 3, mention is made of ``various places in the literature''. It would be nice to know what these places are.

2) On page 21, there is a similar issue with the statement ``We give references below''. There are a handful of references to older papers in what follows, but too much work is left to the reader. It would be nice to have a couple of specific sentences outlining what is new here and what is drawn from the literature, with equation numbers.

3) I noticed a handful of typos, ``indepenent'' on p 11, ``one can be build'' on p 17.

See report

See report

My view is that the authors in the end took a large amount of the advice given by the referees including myself previously, in particular reducing a lot the unnecessary clutter in Sections 3 and 4. The paper is now more streamlined and is easier to follow.

(I did not say that there was absolutely nothing new in Section 4. In fact the authors ended up doing mostly what I wanted, which was to focus on the z\ne 1,2 limits in a more streamlined section.)

I had a typo in my earlier review which was that I wrote to talk about "large z" limit but that should have been "large v" limit, although I suppose both are worthwhile -- since this is clearly my "fault" I would not bother the authors to add this to the paper unless they felt compelled to.

I think the paper can be published as is.

Thank you for the positive recommendation.

## Anonymous on 2018-06-06

(in reply to Report 2 on 2018-06-05)We find the comments of this referee difficult to place in context. Firstly, whether this manuscript reminds the referee of some other journal style, we do not find a very useful comment that we can do something with. Secondly, the referee has a suspicion that most part is not new, but we do not know how to respond to 'suspicion'. The comment about the "newness" of equation (2.10) is a mystery to us, as the velocity is not the gradient of the phase of an order parameter. We do agree though that is interesting to apply our framework to superfluids, as we mention in the discussion and outlook.

As for the requested changes, we believe we have stressed very clearly and explicitly in section 4 what are for sure the new equations, namely the velocity dependent terms in the partition function and thermodynamic quantities, and the speed of sound. We have given the relevant references [32,33] and connected them to e.g. the relevant equations. Some results where known for z=2, but the other referee asked us to take out this special case. Still reference [33] is mentioned connected to eqn (4.45). Whether e.g. the formula for the heat capacity in eqn (4.12) is new, we don't know and we make no strong claims there. We have not found it anywhere in the literature, but agree it is a straightforward generalisation, as we write in the beginning of section 4.

We would be happy to correct the typos the referee mentions in a later stage.