Dynamic Scaling of Two-Dimensional Polar Flocks
Abstract: We propose a hydrodynamic description of the homogeneous ordered phase of polar flocks. Starting from symmetry principles, we construct the appropriate equation for the dynamics of the Goldstone mode associated with the broken rotational symmetry. We then focus on the two-dimensional case considering both "Malthusian flocks" for which the density field is a fast variable that does not enter the hydrodynamic description and "Vicsek flocks" for which it does. In both cases, we argue in favor of scaling relations that allow to compute exactly the scaling exponents, which are found in excellent agreement with previous simulations of the Vicsek model and with the numerical integration of our hydrodynamic equations.
- J. Toner and Y. Tu, Long-range order in a two-dimensional dynamical XY model: how birds fly together, Physical Review Letters 75, 4326 (1995).
- B. Mahault, F. Ginelli, and H. Chaté, Quantitative assessment of the Toner and Tu theory of polar flocks, Physical Review Letters 123, 218001 (2019), publisher: APS.
- H. Chaté, Dry aligning dilute active matter, Annual Review of Condensed Matter Physics 11, 189 (2020), publisher: Annual Reviews.
- N. D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic Heisenberg models, Physical Review Letters 17, 1133 (1966).
- P. Hohenberg, Existence of long-range order in one and two dimensions, Physical Review 158, 383 (1967).
- J. Toner, Reanalysis of the hydrodynamic theory of fluid, polar-ordered flocks, Physical Review E 86, 031918 (2012a).
- M. Besse, H. Chaté, and A. Solon, Metastability of Constant-Density Flocks, Physical Review Letters 129, 268003 (2022), publisher: APS.
- L. Chen, C. F. Lee, and J. Toner, Universality class for a nonequilibrium state of matter, Physical Review E 102, 022610 (2020a), publisher: APS.
- L. Di Carlo and M. Scandolo, Evidence of fluctuation-induced first-order phase transition in active matter, New Journal of Physics 24, 123032 (2022), publisher: IOP Publishing.
- Y. Tu, J. Toner, and M. Ulm, Sound waves and the absence of Galilean invariance in flocks, Physical review letters 80, 4819 (1998).
- N. Kyriakopoulos, F. Ginelli, and J. Toner, Leading birds by their beaks: the response of flocks to external perturbations, New Journal of Physics 18, 073039 (2016), publisher: IOP Publishing.
- The large crossover scale at small noise explains why we erroneously concluded before [10] that the scaling of fluctuations in Malthusian flocks were consistent with the predictions of Ref. [9].
- We could in principle complement Eq. (13) with a conserved noise term but it is always irrelevant compared to the non-conserved noise on θ𝜃\thetaitalic_θ.
- D. Geyer, A. Morin, and D. Bartolo, Sounds and hydrodynamics of polar active fluids, Nature materials 17, 789 (2018), publisher: Nature Publishing Group.
- H. Ikeda, How advection stabilize long-range order in two dimensions, arXiv preprint arXiv:2401.01603 (2024).
- P. Jentsch and C. F. Lee, A new universality class describes Vicsek’s flocking phase in physical dimensions, arXiv preprint arXiv:2402.01316 (2024).
- B. Delamotte, An introduction to the nonperturbative renormalization group (Springer, 2012) pp. 49–132.
- D. Forster, D. R. Nelson, and M. J. Stephen, Large-distance and long-time properties of a randomly stirred fluid, Physical Review A 16, 732 (1977).
- L. Canet, B. Delamotte, and N. Wschebor, Fully developed isotropic turbulence: Symmetries and exact identities, Physical Review E 91, 053004 (2015), publisher: APS.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.