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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
Submission summary
| Authors (as registered SciPost users): | Kevin Grosvenor · Niels Obers |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2501.00025v3 (pdf) |
| Date accepted: | Sept. 2, 2025 |
| Date submitted: | Aug. 13, 2025, 3:09 p.m. |
| Submitted by: | Kevin Grosvenor |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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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
Author comments upon resubmission
List of changes
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Page 3: Promoted what used to be footnote 2 (origin of the term boid) to the main text.
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New footnote 7 states that "collision time" approximation may also be called "relaxation time" approximation, but that this does not just mean low number density, but also depends on the interaction strength.
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Augmented what used to be footnote 9 to clarify the enslavement procedure to terminate the Vlasov hierarchy.
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Page 9: Comment on the standard expression for the equilibrium expectation value of the product of two velocities. This holds in the bulk, away from boundaries.
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Page 10: Added clarifying statements about the influence kernel - it captures only the inter-boid interactions, which are assumed to be translation-invariant and isotropic.
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Added footnote 15 stating the assumption of the analyticity of the small-v expansion.
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Page 14: Added a statement that the renormalization group analysis of Toner-Tu theory is not fully resolved and cite the four recent relevant works.
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Page 14: Added a statement that noise can be added by hand, as is done in Toner-Tu theory.
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Page 16: In response to Referee #3, added a discussion about work by Thomas Ihle deriving Toner-Tu theory from a coarse-graining of the Vicsek model.
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Page 23: Added a Method 2 for how to solve the problem of modifying the influence kernel to generate diffusion coefficients while remaining well-defined in real space. This was inspired by the question from Referee #3 about Hessian terms in the influence kernel and so we thank the referee in footnote 27.
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Page 28: Corrected two typos: 1) the numerator of m_{in} in (E3) should be m, not 1; and 2) the term (v.nabla)rho in (E5) should be nabla.(rho v).
Published as SciPost Phys. 19, 071 (2025)