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Hydrodynamics without Boost-Invariance from Kinetic Theory: From Perfect Fluids to Active Flocks
by Kevin T. Grosvenor, Niels A. Obers, Subodh P. Patil
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Kevin Grosvenor · Niels Obers |
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| Preprint Link: | https://arxiv.org/abs/2501.00025v2 (pdf) |
| Date submitted: | Jan. 16, 2025, 4:23 a.m. |
| Submitted by: | Kevin Grosvenor |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
We derive the hydrodynamic equations of perfect fluids without boost invariance [1] from kinetic theory. Our approach is to follow the standard derivation of the Vlasov hierarchy based on an a-priori unknown collision functional satisfying certain axiomatic properties consistent with the absence of boost invariance. The kinetic theory treatment allows us to identify various transport coefficients in the hydrodynamic regime. We identify a drift term that effects a relaxation to an equilibrium where detailed balance with the environment with respect to momentum transfer is obtained. We then show how the derivative expansion of the hydrodynamics of flocks can be recovered from boost non-invariant kinetic theory and hydrodynamics. We identify how various coefficients of the former relate to a parameterization of the so-called equation of kinetic state that yields relations between different coefficients, arriving at a symmetry-based understanding as to why certain coefficients in hydrodynamic descriptions of active flocks are naturally of order one, and others, naturally small. When inter-particle forces are expressed in terms of a kinetic theory influence kernel, a coarse-graining scale and resulting derivative expansion emerge in the hydrodynamic limit, allowing us to derive diffusion terms as infrared-relevant operators distilling different parameterizations of microscopic interactions. We conclude by highlighting possible applications.
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
Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 3) on 2025-6-18 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2501.00025v2, delivered 2025-06-18, doi: 10.21468/SciPost.Report.11427
Strengths
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Original results
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Clearly written.
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Of interest to several communities.
Weaknesses
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Comparison with existing literature not thorough enough
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Some of the main results may not be especially novel.
Report
The main ingredient is the introduction of a kinetic mass density that determines a relation between velocity and momentum. There are some assumptions concerning the expectation values of product of operators that are introduced to simplify the kinetic theory derivation, but that seem reasonable and on par with related literature.
Overall the paper looks very solid technically and presents new results, so in my opinion it should be published. There are just some of questions/comments that I think should be addressed by the authors before.
Requested changes
1) In the original Toner and Tu paper (ref [20]), the hydrodynamic equations (1) in their paper take the form in equation (50) with the values for the coefficients (53). There is a foonote ([6]) stating that $\lambda_1\neq 1$ and $\lambda_2\neq 0$ are irrelevant in the IR. This seems to be a very similar statement to the one the authors are making in the paper. In the Toner and Tu paper the demonstration is deferred to another publication. The authors probably should mention this and check if such derivaiton exists, and cite it accordingly if it is the case.
2) This referencehttps://arxiv.org/abs/1006.1825v2 derives Toner Tu equations from Vicsek model using kinetic theory. It is probably worth comparing its results with the results of this paper and checking if there are other similar works.
3) In equation (37) rotational invariance would also allow for a contribution to the kernel of the form $K^i_j=\partial^i \partial_j \phi$, with $\phi$ a scalar function. Depending on what $\phi$ is, this is not necessarily suppressed in the derivative expansion. How would this affect to the discussion? A related question is whether parity invariance is assumed in the whole derivation, or if it could be broken without introducing new terms.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report #2 by Anonymous (Referee 2) on 2025-4-15 (Contributed Report)
- Cite as: Anonymous, Report on arXiv:2501.00025v2, delivered 2025-04-15, doi: 10.21468/SciPost.Report.11018
Strengths
This paper addresses research directions that are both fundamental and applied.
1- On the conceptual side, it highlights the fact that most systems break boost symmetries, making it necessary to develop non-boost-invariant theories—precisely the aim of this work.
2- The paper also establishes connections with well-established theories of flocking hydrodynamics, refining our understanding of these models.
3- The mathematical developments are clearly presented, with many technical details provided in the appendices. Both the results and the outlook are insightful and thoroughly discussed.
Weaknesses
Report
From the hydrodynamic equations, the authors:
1 - Successfully recover the non-boost-invariant hydrodynamics of an ideal fluid by setting the boid forces to zero and imposing detailed balance (i.e., a vanishing collision functional).
2 - Discuss the case where boid forces tend to align particle velocities with a local average velocity field, inspired by the Vicsek model.
3 - Recover the Toner-Tu theory by adopting the Bhatnagar-Gross-Krook-Welander (BGKW) form for the collision functional, setting the boid forces to zero, and introducing the notion of an equation of kinetic state to effectively capture the relation between canonical momentum and velocity. This top-down approach allows the authors to extract information about the parameters of the Toner-Tu theory and to provide symmetry-based explanations for the technical naturalness of some of these parameters.
Requested changes
This work is detailed, compelling, and insightful. Before recommending it for publication, I would like to raise a few questions, comments, and suggestions:
1 - Below equation (11), it is stated that the interactions with the environment are included in the collision functional of Boltzmann’s equation. It is then mentioned that the BGKW expression will be used for the collision functional. To the best of my knowledge, the BGKW expression is based on the physical interpretation that the collision term describes the rate at which collisions change the distribution function over time. The explicit BGKW form assumes that the duration of collisions is much shorter than the relaxation time. For inter-particle interactions, this corresponds to requiring that the interaction range is much smaller than the typical inter-particle distance (i.e., a dilute gas of “little balls”). What, then, are the requirements on the interactions with the environment for the BGKW approximation to remain valid?
2 - Related to question 1: below equation (13), $\tau$ is referred to as a “collision time.” Shouldn’t it instead be interpreted as a relaxation time—the typical time between “instantaneous” collisions?
3 - In equation (30), the most general expression is given without gradients. Is the reason for excluding gradients that the left-hand side corresponds to an equilibrium (ideal fluid) situation?
4 - Below equation (36), the phrase “when supplemented with spatial translation invariance of the system” is used. However, due to the presence of interactions with the environment, the system exchanges momentum with its surroundings. I would therefore naively expect that spatial translation is not a symmetry of the system, since momentum is not conserved. Should this statement instead refer specifically to the spatial translational invariance of the boid forces?
5 - Throughout the paper, isotropy is used to argue that certain quantities—particularly the kinetic mass density—scale as $v^2$. Why couldn’t they instead depend on the norm of the velocity?
6 - (Optional) For clarity, it would be helpful to either add a reference or provide a brief physical explanation for why truncating the Vlasov hierarchy at third order in velocity momenta is a good approximation.
7 - (Optional) To improve accessibility, it may be beneficial to define the specific vocabulary of flocking theory in the Introduction, in order to prevent non-specialist readers from misinterpreting key concepts. For example, in the Introduction, just below equation (4), the phrase “where inter-boid forces are presumed to vanish” could be misleading for non-specialists, as inter-boid forces might be naively interpreted as collisions between particles. This could cause confusion, since equation (4) still contains a (BGKW) collision term. That said, the meaning becomes clear once the full set of computations is presented.
Recommendation
Ask for minor revision