Hamiltonian structure of 2D fluid dynamics with broken parity

The authors identify new conditions on odd parity transport coefficients in 2D for the resulting hydrodynamic equations to be (i) energy conserving (ii) Hamiltonian. The results are very interesting and should certainly be published in Scipost. There were a few points that I would like the authors to clarify before publication:

Authors: We would like to thank the referee for carefully going over the manuscript and recommending it for publication. The referee raised useful questions that helped us to improve the manuscript. Below are the responses to the specific questions raised by the referee.

1) In determining when these systems are Hamiltonian (the criterion for energy conservation is reasonably unambiguous), the authors assume that (i) the Hamiltonian is the energy (ii) the brackets take the specific form defined by Eqs. (18), (19), (26). Within these assumptions, they derive conditions on the transport coefficients for the system to be Hamiltonian. As far as I can tell, these are sufficient conditions – can the authors rule out the possibility of a different Hamiltonian (i.e. not necessarily the energy) and different Poisson brackets whenever the conditions of Statement III are not met? Or does the authors’ physical interpretation of Statement II show that it captures all possible Hamiltonian cases? As things are written, the claim seems to be a bit vague, e.g. on p3, “we will address when the energy conserving fluid dynamics described in Statement I can be endowed with the Hamiltonian structure”, “we show that not all cases in Statement I can possess Hamiltonian structure”, and Statement III makes no reference to a particular choice of bracket.

Authors: Although we have not explicitly worked out the necessary conditions for such, in the manuscript, we considered that:

I. We must recover the ideal fluid Hamiltonian and Poisson algebra in the limit when the viscosity tensor vanishes;

II. The algebra must be local (necessary for uniqueness of the brackets).

These are fairly restrictive and one can believe our results are not only sufficient, but necessary conditions. In fact, assumption I states that to obtain a Hamiltonian system with a different Hamiltonian, we first need to show that there is another conserved quantity that recovers the ideal fluid Hamiltonian in the limit when \eta_{ijkl}\rightarrow0.

We have included these 2 assumptions in the text.

2) The authors mention in the abstract that their results constrain odd-parity hydrodynamics that derives from coarse-grained microscopic systems. It seems to me that this is only true if Eqs. (18), (19), (26) also derive from coarse-graining the microscopic canonical brackets in some way – can this be justified? It seems plausible for Eqs. (18) and (19) but less so for Eq. (26). If not, there is the risk that being Hamiltonian is a formal property of the macroscopic hydrodynamics that is unrelated to the microscopic dynamics.

Authors: If coarse-graining procedure starts from a Hamiltonian system and this process is done consistently, the resulting system should preserve the Hamiltonian structure. In fact, we show in the Statement II that the hydrodynamic fields can be redefined (see Eq. (30)) in such a way that, starting from Eq. (26), we recover the diffeomorphism algebra, i.e., Eq. (20). From a physical point of view, one can always identify these diffeomorphism generators to the density of the center-of-mass molecule momenta (as it has been done in the Ref. [11]).

The starting point of our work is already a hydrodynamic system, which can be viewed as the end product of a consistent coarse-graining procedure.

3) Is there any physical significance to the fact that the “counting scheme” on p6 differs from the usual hydrodynamic derivative expansion? Naively this mixing of different orders in the derivative expansion seems difficult to reconcile with hydrodynamic scaling (e.g. the “Madelung terms” are dispersive and usually neglected in low-order hydrodynamics)

Authors: In this paper, we consider Eqs. (1-4) as our starting point. From here, one can ask if these equations allow for a third conserved quantity, namely the fluid energy. Since we are interested in Hamiltonian systems, the energy must be fully conserved, without neglecting higher-order derivatives. From this viewpoint, this counting scheme is very natural, as can be seen in the Appendix A. Moreover, this is a natural scheme in the Hamiltonian fluid dynamics, as one can see that preserves the Poisson structure (18-20) and (26).

From a more physical ground, such a counting scheme seems to be more appropriate in the non-relativistic hydrodynamics, which allows the fluctuations of the fluid density to be of the same order of the velocity flow. In fact, we will address this in a future work, namely, using the count scheme to determine the most general conditions on the transport coefficients to ensure the non-positive production of fluid energy. It is also an interesting project to understand this counting scheme from the Chapman-Enskog point of view in kinetic theory.

4) The explicit connection to nonholonomic constraints in Sec. IV is nice. Is this expected in all cases that violate Statement III? (there seems to be a claim in the literature that violating Jacobi is always equivalent to a nonholonomic constraint in finite dimensions, e.g. Van Der Schaft, Maschke, Rep. Math. Phys. 34 2 p.225-233, 1994). The way things are currently stated in Sec. IV seems possibly overly general: “In this section, we consider the fluid dynamics described by Case 1 of Statement I, but not satisfying the condition of Statement II. We show that it arises from Hamiltonian fluid dynamics with internal angular momentum degree of freedom subject to a nonholonomic constraint”, given that it is illustrated with one specific example.

Authors: In fact, the failure of the Jacobi identity is the definition of a nonholonomic system, in the Hamiltonian framework (Ref. [27]). We added a sentence in main text stating that.

We were not aware of such reference when writing the paper. We thank the referee to bring it to our attention, we cited this paper in the revised version.

Regarding whether our approach in the Sec. IV is more general than we presented there or not, we believe so. However, since this was supposed to be just an illustrative example, we have not worked it out for the most general case. We will address this method to a larger class of nonholonomic system in a future publication.

5) The conclusion felt a little technical and confusing, with references back to very specific details and some new details added (e.g. central extensions). Please consider revising.

Authors: We agree with the referee on this regard and we removed some of the more technical discussion.

Finally, notation/typos: I found v_i^2 in Eqs. 14-16 a bit confusing, similarly j_i^2 elsewhere. Could the authors consider an alternative notation?

Authors: To the best of our knowledge, this is a fairly standard notation. In this revised version, we did however explain it when this first appears.

2) Appendices: what (four times) -> which

Authors: Fixed.

GMM, AGA and SG.

Listed in the author comments/response.