pith. sign in

arxiv: 2607.02318 · v1 · pith:5FS5N2QTnew · submitted 2026-07-02 · ❄️ cond-mat.soft

Hydrodynamics, Renormalization Group, and Universality Classes Far from Equilibrium

Patrick Jentsch , Chiu Fan Lee This is my paper

Pith reviewed 2026-07-03 04:48 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords nonequilibrium universality classeshydrodynamicsrenormalization groupactive mattercritical phenomenasymmetry principlesliving matter
0
0 comments X

The pith

Hydrodynamic theories built solely from symmetries and conservation laws organize nonequilibrium systems into universality classes without equilibrium counterparts.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper provides a unified review of how universality extends beyond equilibrium by constructing hydrodynamic descriptions from symmetry principles alone. It then applies dynamic renormalization group techniques to classify the resulting theories into distinct universality classes according to their scaling properties. Many of these classes emerge in active and living matter and lack direct equilibrium analogues. The review connects developments since 2015 through shared physical principles rather than isolated examples. A sympathetic reader would value this because it supplies a systematic route to predict large-scale behavior in driven many-body systems.

Core claim

Hydrodynamic equations derived from symmetries and conservation laws alone can be organized by perturbative dynamic renormalization group analysis into genuinely nonequilibrium universality classes, many of which have been identified in active matter since 2015 and exhibit scaling behaviors with no equilibrium counterparts.

What carries the argument

Hydrodynamic theories constructed from symmetry and conservation laws, then classified by dynamic renormalization group flow into universality classes based on scaling behavior.

If this is right

  • Systems sharing the same symmetries and conservation laws fall into the same universality class independent of microscopic constituents.
  • Breaking of equilibrium symmetries such as time-reversal generates new universality classes with distinct scaling exponents.
  • The renormalization group organizes hydrodynamic theories into a finite set of classes whose critical behavior can be computed perturbatively.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same symmetry-based construction could classify driven systems outside active matter, such as certain granular or turbulent flows.
  • Quantitative comparison of measured scaling exponents in living systems against the reviewed classes would test the framework's predictive power.
  • Links between these nonequilibrium classes and known equilibrium ones might emerge from extensions that restore broken symmetries in controlled limits.

Load-bearing premise

Hydrodynamic theories can be systematically constructed from symmetry and conservation laws alone.

What would settle it

An experimental or numerical observation of a driven system whose long-wavelength fluctuations require microscopic details beyond its symmetries and conservation laws to match any predicted universality class would falsify the construction method.

Figures

Figures reproduced from arXiv: 2607.02318 by Chiu Fan Lee, Patrick Jentsch.

Figure 1
Figure 1. Figure 1: Schematic illustration of the concept underlying the Wilsonian renormalization group (RG). Starting from a hydrodynamic theory defined at the microscopic hydrodynamic cutoff scale Λ −1 hydro, fluctuations within a thin momentum shell (yellow highlight) are integrated out, thereby increasing the effective minimum length scale to Λ(ℓ) −1 . This coarse-graining procedure changes the value of the effective dim… view at source ↗
Figure 2
Figure 2. Figure 2: RG flow of the fluctuating incompressible Navier–Stokes equation. Renormalization group (RG) flow of the dimensionless coupling gλ as a function of spatial dimension d, deduced from our perturbative calculation to the one-loop level. The red curve denotes the nontrivial fixed-point value g ∗ λ [Eq. (57)], which exists only below (and strictly speaking, close to) the upper critical dimension dc = 2. For d >… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic RG flow structure of the Kardar-Parisi-Zhang (KPZ) equation. RG flow as a function of spatial dimension d, shown in terms of the dimensionless coupling gKPZ. Grey lines with arrows indicate the corresponding RG flow directions. For d < 2, the perturbative KPZ fixed point (red) is the only stable fixed point and therefore generically governs the large-scale behaviour of the system. In contrast, th… view at source ↗
Figure 4
Figure 4. Figure 4: A “periodic table” of universality classes. As discussed in Section 4.10, the space of UCs is intrinsically high-dimensional. For clarity, we therefore classify the known UCs according to (i) the number and type of hydrodynamic variables involved, and (ii) whether they govern a phase or a critical point. The question mark highlights regions of this space where additional UCs may yet await discovery. 31/41 … view at source ↗
read the original abstract

Universality is one of the central organising principles of modern physics, explaining why systems with vastly different microscopic constituents can exhibit identical large-scale behaviour. While the classification of equilibrium critical phenomena through hydrodynamics and the renormalization group (RG) is now well established, our understanding of universality far from equilibrium remains far less developed. In recent years, however, rapid progress - driven in large part by developments in active and living matter - has uncovered a growing range of genuinely nonequilibrium universality classes (UCs) with no equilibrium counterparts. In this review, we present a pedagogical and unified introduction to hydrodynamic and RG approaches to nonequilibrium many-body systems. We first show how hydrodynamic theories can be systematically constructed from symmetry and conservation laws alone. We then introduce perturbative dynamic RG methods and demonstrate how hydrodynamic theories are organised into distinct UCs according to their scaling behaviour. Building on these foundations, we review the diverse nonequilibrium UCs uncovered since 2015, while emphasizing the conceptual connections and unifying physical principles underlying their emergence. We conclude by discussing open theoretical and experimental challenges for the field.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript is a review article offering a pedagogical introduction to hydrodynamic theories for nonequilibrium many-body systems constructed from symmetry and conservation laws, followed by perturbative dynamic RG methods to classify scaling behavior into universality classes (UCs). It synthesizes post-2015 examples of genuinely nonequilibrium UCs with no equilibrium counterparts, primarily from active and living matter, while highlighting conceptual connections and open challenges.

Significance. If the synthesis is accurate, the review provides a valuable unifying framework for a rapidly advancing field. The explicit emphasis on constructing hydrodynamics from symmetries plus conservation laws, followed by RG organization into UCs, is a pedagogical strength that organizes the literature without introducing new derivations. No machine-checked proofs or code are present, as expected for a review, but the focus on falsifiable scaling predictions in the reviewed examples adds value.

minor comments (1)
  1. [Abstract] Abstract: the phrasing 'systematically constructed from symmetry and conservation laws alone' would benefit from a parenthetical reference to standard caveats in the literature (e.g., closure assumptions in active systems) to avoid implying universality of the procedure.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of its pedagogical strengths, and recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity: review paper with no new derivations

full rationale

This is a review article that synthesizes post-2015 literature on nonequilibrium universality classes. It presents standard methods for constructing hydrodynamic theories from symmetries and conservation laws, followed by dynamic RG, without performing original derivations, fits, or predictions that could reduce to inputs defined within the paper. No load-bearing steps rely on self-citations that themselves depend on the present work, and the central claims rest on external examples rather than internal closure. The paper is self-contained against external benchmarks as a pedagogical summary.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review article; no new free parameters, axioms, or invented entities are introduced by the authors.

pith-pipeline@v0.9.1-grok · 5715 in / 934 out tokens · 39479 ms · 2026-07-03T04:48:11.198541+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

71 extracted references · 63 canonical work pages

  1. [1]

    Hohenberg, P . C. & Halperin, B. I. Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479, DOI: 10.1103/ RevModPhys.49.435 (1977)

    1977

  2. [2]

    & Zhang, Y .-C

    Kardar, M., Parisi, G. & Zhang, Y .-C. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892, DOI: 10.1103/PhysRevLett.56.889 (1986). 33/41

    work page doi:10.1103/physrevlett.56.889 1986

  3. [3]

    & Shochet, O

    Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I. & Shochet, O. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229, DOI: 10.1103/PhysRevLett.75.1226 (1995)

    work page doi:10.1103/physrevlett.75.1226 1995

  4. [4]

    Toner, J. & Tu, Y . Long-range order in a two-dimensional dynamical XY model: How birds fly together. Phys. Rev. Lett. 75, 4326–4329, DOI: 10.1103/PhysRevLett.75.4326 (1995)

    work page doi:10.1103/physrevlett.75.4326 1995

  5. [5]

    & Goldbart, P

    Stone, M. & Goldbart, P . Mathematics for physics (Cambridge University Press, Cambridge, England, 2009)

    2009

  6. [6]

    Kadanoff, L. P . & Martin, P . C. Hydrodynamic Equations and Correlation Functions. Annals Phys. 281, 800–852, DOI: 10.1006/aphy.2000.6023 (2000)

    work page doi:10.1006/aphy.2000.6023 2000

  7. [7]

    Forster, D., Nelson, D. R. & Stephen, M. J. Large-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A 16, 732–749, DOI: 10.1103/PhysRevA.16.732 (1977)

    work page doi:10.1103/physreva.16.732 1977

  8. [8]

    Dupuis, N. et al. The nonperturbative functional renormalization group and its applications. Phys. Reports 910, 1–114, DOI: 10.1016/j.physrep.2021.01.001 (2021)

    work page doi:10.1016/j.physrep.2021.01.001 2021

  9. [9]

    & Kogut, J

    Wilson, K. & Kogut, J. The renormalization group and the ε expansion. Phys. Reports 12, 75–199, DOI: 10.1016/ 0370-1573(74)90023-4 (1974)

    1974

  10. [10]

    Edwards, S. F. & Wilkinson, D. R. The surface statistics of a granular aggregate. Proc. Royal Soc. London. A. Math. Phys. Sci. 381, 17–31, DOI: 10.1098/rspa.1982.0056 (1982). https://royalsocietypublishing.org/rspa/article-pdf/381/1780/17/ 64766/rspa.1982.0056.pdf

    work page doi:10.1098/rspa.1982.0056 1982

  11. [11]

    The non-perturbative sides of the kardarparisizhang equation

    Canet, L. The non-perturbative sides of the kardarparisizhang equation. J. Stat. Mech. Theory Exp. 2025, 124003, DOI: 10.1088/1742-5468/ae1e75 (2025)

    work page doi:10.1088/1742-5468/ae1e75 2025

  12. [12]

    E., Delamotte, B

    Besse, M., Fausti, G., Cates, M. E., Delamotte, B. & Nardini, C. Interface roughening in nonequilibrium phase-separated systems. Phys. Rev. Lett. 130, 187102, DOI: 10.1103/PhysRevLett.130.187102 (2023)

    work page doi:10.1103/physrevlett.130.187102 2023

  13. [13]

    Toner, J., Guttenberg, N. & Tu, Y . Swarming in the dirt: Ordered flocks with quenched disorder. Phys. Rev. Lett. 121, 248002, DOI: 10.1103/PhysRevLett.121.248002 (2018)

    work page doi:10.1103/physrevlett.121.248002 2018

  14. [14]

    Toner, J., Guttenberg, N. & Tu, Y . Hydrodynamic theory of flocking in the presence of quenched disorder. Phys. Rev. E 98, 062604, DOI: 10.1103/PhysRevE.98.062604 (2018)

    work page doi:10.1103/physreve.98.062604 2018

  15. [15]

    Chen, L., Lee, C. F. & Toner, J. Moving, reproducing, and dying beyond flatland: Malthusian flocks in dimensions d > 2. Phys. Rev. Lett. 125, 098003, DOI: 10.1103/PhysRevLett.125.098003 (2020)

    work page doi:10.1103/physrevlett.125.098003 2020

  16. [16]

    Chen, L., Lee, C. F. & Toner, J. Universality class for a nonequilibrium state of matter: A d = 4 ε expansion study of malthusian flocks. Phys. Rev. E 102, 022610, DOI: 10.1103/PhysRevE.102.022610 (2020)

    work page doi:10.1103/physreve.102.022610 2020

  17. [17]

    Birth, death, and horizontal flight: Malthusian flocks with an easy plane in three dimensions

    Toner, J. Birth, death, and horizontal flight: Malthusian flocks with an easy plane in three dimensions. Phys. Rev. E 110, 064604, DOI: 10.1103/PhysRevE.110.064604 (2024)

    work page doi:10.1103/physreve.110.064604 2024

  18. [18]

    Chen, L., Lee, C. F. & Toner, J. Incompressible polar active fluids in the moving phase in dimensions d > 2. New J. Phys. 20, 113035, DOI: 10.1088/1367-2630/aaec31 (2018)

    work page doi:10.1088/1367-2630/aaec31 2018

  19. [19]

    Chen, L., Lee, C. F. & Toner, J. Mapping two-dimensional polar active fluids to two-dimensional soap and one- dimensional sandblasting. Nat. Commun. 7, 12215, DOI: 10.1038/ncomms12215 (2016)

    work page doi:10.1038/ncomms12215 2016

  20. [20]

    F., Maitra, A

    Chen, L., Lee, C. F., Maitra, A. & Toner, J. Dynamics of packed swarms: Time-displaced correlators of two-dimensional incompressible flocks. Phys. Rev. E 109, L012601, DOI: 10.1103/PhysRevE.109.L012601 (2024)

    work page doi:10.1103/physreve.109.l012601 2024

  21. [21]

    F., Maitra, A

    Chen, L., Lee, C. F., Maitra, A. & Toner, J. Packed swarms on dirt: Two-dimensional incompressible flocks with quenched and annealed disorder. Phys. Rev. Lett. 129, 188004, DOI: 10.1103/PhysRevLett.129.188004 (2022)

    work page doi:10.1103/physrevlett.129.188004 2022

  22. [22]

    F., Maitra, A

    Chen, L., Lee, C. F., Maitra, A. & Toner, J. Hydrodynamic theory of two-dimensional incompressible polar active fluids with quenched and annealed disorder. Phys. Rev. E 106, 044608, DOI: 10.1103/PhysRevE.106.044608 (2022)

    work page doi:10.1103/physreve.106.044608 2022

  23. [23]

    P ., Sahoo, S

    Dadhichi, L. P ., Sahoo, S. K., Kumar, K. V . & Ramaswamy, S. Superdiffusion and antidiffusion in an aligned active suspension. Phys. Rev. Lett. 136, 128302, DOI: 10.1103/d61l-1tyh (2026)

    work page doi:10.1103/d61l-1tyh 2026

  24. [24]

    Mahdisoltani, S., Zinati, R. B. A., Duclut, C., Gambassi, A. & Golestanian, R. Nonequilibrium polarity-induced chemotaxis: Emergent galilean symmetry and exact scaling exponents. Phys. Rev. Res. 3, 013100, DOI: 10.1103/ PhysRevResearch.3.013100 (2021)

    2021

  25. [25]

    & Brachet, M

    Cartes, C., Tirapegui, E., Pandit, R. & Brachet, M. The galerkin-truncated burgers equation: crossover from inviscid- thermalized to kardarparisizhang scaling. Philos. Transactions Royal Soc. A: Math. Phys. Eng. Sci. 380, 20210090, DOI: 10.1098/rsta.2021.0090 (2022). https://royalsocietypublishing.org/rsta/article-pdf/doi/10.1098/rsta.2021.0090/1322947/ r...

    work page doi:10.1098/rsta.2021.0090 2021

  26. [26]

    N., Castro, M

    Rodríguez-Fernández, E., Santalla, S. N., Castro, M. & Cuerno, R. Anomalous ballistic scaling in the tensionless or inviscid kardar-parisi-zhang equation. Phys. Rev. E 106, 024802, DOI: 10.1103/PhysRevE.106.024802 (2022)

    work page doi:10.1103/physreve.106.024802 2022

  27. [27]

    & Canet, L

    Fontaine, C., V ercesi, F., Brachet, M. & Canet, L. Unpredicted scaling of the one-dimensional kardar-parisi-zhang equation. Phys. Rev. Lett. 131, 247101, DOI: 10.1103/PhysRevLett.131.247101 (2023)

    work page doi:10.1103/physrevlett.131.247101 2023

  28. [28]

    & Canet, L

    Gosteva, L., Tarpin, M., Wschebor, N. & Canet, L. Inviscid fixed point of the multidimensional burgers–kardar-parisi- zhang equation. Phys. Rev. E 110, 054118, DOI: 10.1103/PhysRevE.110.054118 (2024)

    work page doi:10.1103/physreve.110.054118 2024

  29. [29]

    Wittkowski, R. et al. Scalar ϕ 4 field theory for active-particle phase separation. Nat. Commun. 5, 4351, DOI: 10.1038/ ncomms5351 (2014)

    2014

  30. [30]

    & Cates, M

    Tjhung, E., Nardini, C. & Cates, M. E. Cluster phases and bubbly phase separation in active fluids: Reversal of the ostwald process. Phys. Rev. X 8, 031080, DOI: 10.1103/PhysRevX.8.031080 (2018)

    work page doi:10.1103/physrevx.8.031080 2018

  31. [31]

    Athermal phase separation of self-propelled particles with no alignment , year =

    Fily, Y . & Marchetti, M. C. Athermal phase separation of self-propelled particles with no alignment. Phys. Rev. Lett. 108, 235702, DOI: 10.1103/PhysRevLett.108.235702 (2012)

    work page doi:10.1103/physrevlett.108.235702 2012

  32. [32]

    Cates, M. E. & Tailleur, J. Motility-induced phase separation. Annu. Rev. Condens. Matter Phys. 6, 219–244, DOI: https://doi.org/10.1146/annurev-conmatphys-031214-014710 (2015)

    work page doi:10.1146/annurev-conmatphys-031214-014710 2015

  33. [33]

    A., Zwicker, D., Jülicher, F

    Weber, C. A., Zwicker, D., Jülicher, F. & Lee, C. F. Physics of active emulsions. Reports on Prog. Phys. 82, 064601– 064601, DOI: 10.1088/1361-6633/ab052b (2019)

    work page doi:10.1088/1361-6633/ab052b 2019

  34. [34]

    Lee, C. F. Interface stability, interface fluctuations, and the gibbsthomson relationship in motility-induced phase separa- tions. Soft Matter 13, 376–385, DOI: 10.1039/C6SM01978A (2017)

    work page doi:10.1039/c6sm01978a 2017

  35. [35]

    Toner, J. & Tu, Y . Flocks, herds, and schools: A quantitative theory of flocking. Phys. Rev. E 58, 4828–4858, DOI: 10.1103/PhysRevE.58.4828 (1998)

    work page doi:10.1103/physreve.58.4828 1998

  36. [36]

    Reanalysis of the hydrodynamic theory of fluid, polar-ordered flocks

    Toner, J. Reanalysis of the hydrodynamic theory of fluid, polar-ordered flocks. Phys. Rev. E 86, 031918, DOI: 10.1103/ PhysRevE.86.031918 (2012)

    2012

  37. [37]

    & Lee, C

    Jentsch, P . & Lee, C. F. New universality class describes vicsek’s flocking phase in physical dimensions. Phys. Rev. Lett. 133, 128301, DOI: 10.1103/PhysRevLett.133.128301 (2024)

    work page doi:10.1103/physrevlett.133.128301 2024

  38. [38]

    & Solon, A

    Chaté, H. & Solon, A. Dynamic scaling of two-dimensional polar flocks. Phys. Rev. Lett. 132, 268302, DOI: 10.1103/ PhysRevLett.132.268302 (2024)

    2024

  39. [39]

    Chen, L. et al. The inconvenient truth about flocks (2025). 2503.17064

    work page arXiv 2025

  40. [40]

    & Toner, J

    Miller, M. & Toner, J. Phase separation in ordered polar active fluids: A completely different universality class from that of equilibrium fluids. Phys. Rev. E 110, 054607, DOI: 10.1103/PhysRevE.110.054607 (2024)

    work page doi:10.1103/physreve.110.054607 2024

  41. [41]

    & Lee, C

    Wong, M. & Lee, C. F. New universality classes govern the critical and multicritical behavior of an active ising model (2025). 2507.06068

    work page arXiv 2025

  42. [42]

    Solon, A. P . & Tailleur, J. Revisiting the flocking transition using active spins. Phys. Rev. Lett. 111, 078101, DOI: 10.1103/PhysRevLett.111.078101 (2013)

    work page doi:10.1103/physrevlett.111.078101 2013

  43. [43]

    Solon, A. P . & Tailleur, J. Flocking with discrete symmetry: The two-dimensional active ising model. Phys. Rev. E 92, 042119, DOI: 10.1103/PhysRevE.92.042119 (2015)

    work page doi:10.1103/physreve.92.042119 2015

  44. [44]

    & Lee, C

    Nesbitt, D., Pruessner, G. & Lee, C. F. Uncovering novel phase transitions in dense dry polar active fluids using a lattice Boltzmann method. New J. Phys. 23, 043047–043047, DOI: 10.1088/1367-2630/abd8c0 (2021)

    work page doi:10.1088/1367-2630/abd8c0 2021

  45. [45]

    Birth, death, and flight: A theory of malthusian flocks

    Toner, J. Birth, death, and flight: A theory of malthusian flocks. Phys. Rev. Lett. 108, 088102, DOI: 10.1103/PhysRevLett. 108.088102 (2012)

    work page doi:10.1103/physrevlett 2012

  46. [46]

    & Lee, C

    Chen, L., Toner, J. & Lee, C. F. Critical phenomenon of the orderdisorder transition in incompressible active fluids. New J. Phys. 17, 042002, DOI: 10.1088/1367-2630/17/4/042002 (2015)

    work page doi:10.1088/1367-2630/17/4/042002 2015

  47. [47]

    Zinati, R. B. A., Besse, M., Tarjus, G. & Tissier, M. Dense polar active fluids in a disordered environment. Phys. Rev. E 105, 064605, DOI: 10.1103/PhysRevE.105.064605 (2022)

    work page doi:10.1103/physreve.105.064605 2022

  48. [48]

    Cavagna, A. et al. Natural swarms in 3.99 dimensions. Nat. Phys. 19, 1043–1049, DOI: 10.1038/s41567-023-02028-0 (2023)

    work page doi:10.1038/s41567-023-02028-0 2023

  49. [49]

    van der Kolk, J. et al. Anomalous collective dynamics of autochemotactic populations. Phys. Rev. Lett. 131, 088201, DOI: 10.1103/PhysRevLett.131.088201 (2023). 35/41

    work page doi:10.1103/physrevlett.131.088201 2023

  50. [50]

    P ., Rosch, A

    Daviet, R., Zelle, C. P ., Rosch, A. & Diehl, S. Nonequilibrium criticality at the onset of time-crystalline order. Phys. Rev. Lett. 132, 167102, DOI: 10.1103/PhysRevLett.132.167102 (2024)

    work page doi:10.1103/physrevlett.132.167102 2024

  51. [51]

    F., Maitra, A

    Chen, L., Lee, C. F., Maitra, A. & Toner, J. Incompressible Polar Active Fluids with Quenched Random Field Disorder in Dimensions d > 2. Phys. Rev. Lett. 129, 198001, DOI: 10.1103/PhysRevLett.129.198001 (2022)

    work page doi:10.1103/physrevlett.129.198001 2022

  52. [52]

    & Ramaswamy, S

    Aditi Simha, R. & Ramaswamy, S. Hydrodynamic Fluctuations and Instabilities in Ordered Suspensions of Self-Propelled Particles. Phys. Rev. Lett. 89, 058101–058101, DOI: 10.1103/PhysRevLett.89.058101 (2002)

    work page doi:10.1103/physrevlett.89.058101 2002

  53. [53]

    Keller, E. F. & Segel, L. A. Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415, DOI: https://doi.org/10.1016/0022-5193(70)90092-5 (1970)

    work page doi:10.1016/0022-5193(70)90092-5 1970

  54. [54]

    & Golestanian, R

    Gelimson, A. & Golestanian, R. Collective dynamics of dividing chemotactic cells. Phys. Rev. Lett. 114, 028101, DOI: 10.1103/PhysRevLett.114.028101 (2015)

    work page doi:10.1103/physrevlett.114.028101 2015

  55. [55]

    & Golestanian, R

    Ben Alì Zinati, R., Duclut, C., Mahdisoltani, S., Gambassi, A. & Golestanian, R. Stochastic dynamics of chemotactic colonies with logistic growth. Europhys. Lett. 136, 50003, DOI: 10.1209/0295-5075/ac48c9 (2022)

    work page doi:10.1209/0295-5075/ac48c9 2022

  56. [56]

    Cardy, J. L. & Sugar, R. L. Directed percolation and reggeon field theory. J. Phys. A: Math. Gen. 13, L423, DOI: 10.1088/0305-4470/13/12/002 (1980)

    work page doi:10.1088/0305-4470/13/12/002 1980

  57. [57]

    & Lee, C

    Legrand, G. & Lee, C. F. Universal behavior at the Lifshitz points of an active Malthusian Ising model. Newton 100563, DOI: 10.1016/j.newton.2026.100563 (2026)

    work page doi:10.1016/j.newton.2026.100563 2026

  58. [58]

    T., Gorshkov, A

    Y oung, J. T., Gorshkov, A. V ., Foss-Feig, M. & Maghrebi, M. F. Nonequilibrium fixed points of coupled ising models. Phys. Rev. X 10, 011039, DOI: 10.1103/PhysRevX.10.011039 (2020)

    work page doi:10.1103/physrevx.10.011039 2020

  59. [59]

    T., Gorshkov, A

    Y oung, J. T., Gorshkov, A. V . & Maghrebi, M. Nonequilibrium universality of the nonreciprocally coupled O ( n 1 ) ΠO ( n 2 ) model. Phys. Rev. E 113, 034132, DOI: 10.1103/n4l9-p6xn (2026)

    work page doi:10.1103/n4l9-p6xn 2026

  60. [60]

    & Lee, C

    Jentsch, P . & Lee, C. F. Critical phenomena in compressible polar active fluids: Dynamical and functional renormalization group studies. Phys. Rev. Res. 5, 023061, DOI: 10.1103/PhysRevResearch.5.023061 (2023)

    work page doi:10.1103/physrevresearch.5.023061 2023

  61. [61]

    Cavagna, A. et al. Scale-free correlations in starling flocks. Proc. Natl. Acad. Sci. 107, 11865–11870, DOI: 10.1073/pnas. 1005766107 (2010)

    work page doi:10.1073/pnas 2010

  62. [62]

    Attanasi, A. et al. Finite-Size Scaling as a Way to Probe Near-Criticality in Natural Swarms. Phys. Rev. Lett. 113, 238102–238102 (2014)

    2014

  63. [63]

    Cavagna, A. et al. Dynamic scaling in natural swarms. Nat. Phys. 13, 914–918, DOI: 10.1038/nphys4153 (2017)

    work page doi:10.1038/nphys4153 2017

  64. [64]

    & Bartolo, D

    Bricard, A., Caussin, J.-B., Desreumaux, N., Dauchot, O. & Bartolo, D. Emergence of macroscopic directed motion in populations of motile colloids. Nature 503, 95–98, DOI: 10.1038/nature12673 (2013)

    work page doi:10.1038/nature12673 2013

  65. [65]

    & Bartolo, D

    Geyer, D., Martin, D., Tailleur, J. & Bartolo, D. Freezing a Flock: Motility-Induced Phase Separation in Polar Active Liquids. Phys. Rev. X 9, 031043–031043, DOI: 10.1103/PhysRevX.9.031043 (2019)

    work page doi:10.1103/physrevx.9.031043 2019

  66. [66]

    Sanchez, T., Chen, D. T. N., DeCamp, S. J., Heymann, M. & Dogic, Z. Spontaneous motion in hierarchically assembled active matter. Nature 491, 431–434, DOI: 10.1038/nature11591 (2012)

    work page doi:10.1038/nature11591 2012

  67. [67]

    J., Redner, G

    DeCamp, S. J., Redner, G. S., Baskaran, A., Hagan, M. F. & Dogic, Z. Orientational order of motile defects in active nematics. Nat. Mater. 14, 1110–1115, DOI: 10.1038/nmat4387 (2015)

    work page doi:10.1038/nmat4387 2015

  68. [68]

    Keber, F. C. et al. Topology and dynamics of active nematic vesicles. Science 345, 1135–1139, DOI: 10.1126/science. 1254784 (2014)

    work page doi:10.1126/science 2014

  69. [69]

    & Trepat, X

    Alert, R. & Trepat, X. Physical Models of Collective Cell Migration. Annu. Rev. Condens. Matter Phys. 11, 77–101, DOI: 10.1146/annurev-conmatphys-031218-013516 (2020)

    work page doi:10.1146/annurev-conmatphys-031218-013516 2020

  70. [70]

    & Shibata, T

    Hayakawa, M., Bhattacherjee, B., Kuwayama, H. & Shibata, T. Emergence of chiral multi-armed spirals in an open system of migrating cells under continuous cell supply (2025). 2511.16074

    work page arXiv 2025

  71. [71]

    Chen, L., Lee, C. F. & Toner, J. Only the ambidextrous can flock: Two-dimensional chiral malthusian flocks, time cholesterics, and the kardar-parisi-zhang equation. Phys. Rev. E 113, 045419, DOI: 10.1103/sj5y-8dsd (2026). 36/41 A Graphical Corrections We present here the 1-loop calculations of the remaining couplings for the Navier-Stokes equation not shown...

    work page doi:10.1103/sj5y-8dsd 2026