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Hydrodynamic Theory of Two-dimensional Chiral Malthusian Flocks

T0 review · 0 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Any nonzero chirality erases long-range orientational order in two-dimensional dry Malthusian flocks, because their fluctuations obey the (2+1)-dimensional KPZ equation.

desk verdict The central KPZ mapping is right, and the paper is honest about its weaker parts; the small-chirality hierarchy is plausible but heuristic. read the letter →

arxiv 2507.17762 v2 pith:YW5EDT4K submitted 2025-07-08 cond-mat.soft cond-mat.stat-mech

Leiming Chen , Chiu Fan Lee , John Toner This is my paper
classification cond-mat.softcond-mat.stat-mech
keywords chiralactivematterMalthusianflocksKardar-Parisi-ZhanguniversalityclasstimecholestericorientationalorderEdwards-Wilkinsonregimehydrodynamictheorytwo-dimensional
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether the perfect rotating order of a two-dimensional chiral dry Malthusian flock—particles that align, reproduce and die, and move over a substrate—survives noise. It argues that fluctuations of the phase of the rotating velocity field obey the (2+1)-dimensional Kardar-Parisi-Zhang (KPZ) equation, whose fluctuations grow without bound. Because of that mapping, any nonzero chirality destroys long-ranged orientational order in the thermodynamic limit. For weak chirality, however, a hierarchy of length and time scales produces an intermediate Edwards-Wilkinson regime with quasi-long-range order and algebraically decaying correlations, followed by nonlinear KPZ and vortex-unbinding regimes. The result predicts specific, measurable velocity and density correlations.

What carries the argument

The load-bearing object is the phase field $φ(r,t)$ defined through $v = v_0[\cos(bt+φ)\hat{x} - \sin(bt+φ)\hat{y}]$; it is the Goldstone mode of the rotating state. After cycle-averaging over the fast rotation, all chiral terms in the velocity equation reduce to the KPZ nonlinearity $(∇φ)^2$, while achiral dissipative terms produce $∇^2φ$, giving the KPZ equation for $φ$. The small-chirality analysis then treats each Fourier mode as a mode of the achiral flock with rotating wavevector components $q_x(t)=q\cos bt$ and $q_y(t)=-q\sin bt$, and averages the damping rate over a cycle; this yields the crossover scales $L_c$, $L_{NL}$, $L_v$ and the exponent relations $η_ν = 2/z_a - 1$, $η_g = 6/z_a - 1$. The compactness of $φ$ (periodic identification $φ \to φ+2πn$) additionally allows vortices, which set the ultimate disorder scale $L_v$.

What would settle it

Simulate a two-dimensional chiral dry Malthusian flock at small chirality $b$ and measure the equal-time velocity correlation $C(r)=\langle v(r,t)\cdot v(0,t)\rangle$ over sizes exceeding $L_c$. The paper predicts $C(r) \sim r^{-α}$ with $α \to 0$ as $b \to 0$ but no plateau at any nonzero $b$; observing a long-range-order plateau $C(\infty)>0$ at fixed $b>0$, or a correlation length that fails to diverge as $b^{-4/5}$, would falsify the mapping.

Watch

Extended reading notes

Core claim

The central claim is that a generic two-dimensional chiral dry Malthusian flock in its rotating “time-cholesteric” state is described hydrodynamically by the phase field $φ(r,t)$ with equation of motion $∂_tφ = ν∇^2φ + (λ_K/2)(∇φ)^2 + f_φ$, a (2+1)-dimensional KPZ equation. Since KPZ fluctuations in 2+1 dimensions diverge with separation, the equal-time velocity correlation decays—eventually becoming short-ranged—so chirality destroys the long-range order that achiral Malthusian flocks possess. In the weak-chirality limit the paper derives explicit scaling laws: the crossover length $L_c \propto b^{-1/z_a}$ with $z_a \approx 5/4$, the nonlinear length $L_{NL} \propto \exp(C_{NL} b^{-19/5})$, the vortex length $L_v \propto \exp(C_{vL} b^{-22/5})$, and an exponent $α = D_φ/(2πν) \propto b^{3/5}$ controlling algebraic decay in the linear regime. The paper also predicts density correlations that grow algebraically with separation and a temporal Fourier spectrum of velocity correlations with power-law broadened peaks at frequencies $±b$.

Load-bearing premise

The central premise is that in the small-chirality limit each Fourier mode of the rotating background decays at the time-averaged rate of the corresponding mode of the achiral flock, with wavevector rotating as $(q\cos bt, -q\sin bt)$; if rotation-induced mode coupling invalidates this averaging, the predicted crossover scales $L_c \propto b^{-4/5}$ and the entire hierarchy would shift.

Editorial extensions

If this is right

  • Any nonzero chirality $b$ eliminates long-range orientational order in two-dimensional dry Malthusian flocks; the average velocity vanishes in the infinite-size limit at fixed $b$.
  • For weak chirality, there is a wide intermediate linear regime $L_c \ll r \ll L_{NL}$ with quasi-long-range order and algebraic velocity correlations $r^{-α}$, with $α \propto b^{3/5}$; this is the regime accessible to simulations and experiments.
  • The temporal Fourier transform of velocity correlations shows power-law broadened peaks at frequencies $±b$, with divergence $|δω|^{α/2-1}$ in a wide frequency range, a signal analogous to Bragg peaks in smectics.
  • Density correlations grow algebraically with separation, with exponent $4χ-4 \approx -2.448$ in the nonlinear KPZ regime and $r^{-4}$ in the linear regime, ruling out giant number fluctuations.
  • Vortices in the compact KPZ phase become important at an exponentially large length $L_v \propto \exp(-C b^{-22/5})$, beyond which the flock is fully disordered.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the same KPZ mapping should apply to other dry active systems with a single broken continuous time-translation symmetry and a uniform rotating state, making the predicted fluctuation universality broader than the specific flock model studied here.
  • Editorial extension: the predicted 'Bragg peaks' at $±b$ in the temporal Fourier spectrum can be probed directly in experiments on chiral colloidal rollers or circularly swimming bacteria; measuring the exponent $α$ as a function of chirality would test the $b^{3/5}$ law.
  • Editorial extension: the paper's vortex analysis is explicitly flagged as speculative; a numerical study of vortex unbinding in the (2+1)-dimensional compact KPZ equation would decide whether $L_v$ really diverges as $α^{-\eta_y}$ with $η_y \approx 22/5$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper studies two-dimensional chiral dry Malthusian flocks, i.e., polar-ordered active matter with broken chiral symmetry and no conservation of momentum or particle number. In the noiseless limit the authors identify a uniformly rotating “time cholesteric” state in which the mean velocity rotates at frequency b. Allowing the phase of this rotation to fluctuate, they project the hydrodynamic velocity equation transverse to the local velocity and average over one rotation cycle, obtaining an equation of motion for the phase that is exactly the (2+1)-dimensional KPZ equation, ∂tφ = ν∇²φ + (λ_K/2)(∇φ)² + f_φ. A symmetry-based enumeration in Appendix A is used to argue that this mapping is generic, i.e., independent of the many terms omitted from the truncated starting model. The paper then derives the enslaved density response, the phase/velocity/density correlations in the linear and nonlinear KPZ regimes, the weak-chirality crossover scales L_c, L_NL, τ_NL and ν(b)∝b^{-3/5} using exponents from simulations of achiral Malthusian flocks, temporal Bragg peaks in the velocity correlation spectrum, and a speculative vortex scale L_v based on an XY analogy for the compact phase. The headline conclusion is that any nonzero chirality destroys long-ranged orientational order in the hydrodynamic limit; for weak chirality, an extended linear (Edwards-Wilkinson) regime gives quasi-long-ranged order on accessible scales.

Significance. If correct, this is an important and surprising result: it places generic 2D chiral Malthusian flocks in the KPZ universality class, which immediately implies short-ranged equal-time velocity correlations and hence destruction of long-range order for any b>0. The central mapping is supported by a model-independent symmetry argument (Appendix A), which is a genuine strength: the conclusion does not depend on the detailed values of the exponents or on the heuristic crossover calculation. The paper is also commendably explicit about the speculative status of the vortex analysis (Section XI) and about the reliance on external simulations for the achiral exponents. The predictions—algebraic decay in the linear regime, stretched-exponential decay in the nonlinear regime, power-law Bragg-peak tails in the temporal spectrum, and the exponential divergence of L_NL as b→0—are concrete and falsifiable. The weakest part is the quantitative small-chirality hierarchy, which rests on a cycle-averaging argument in Section VII; however, this does not affect the central KPZ mapping, since even a massless Edwards-Wilkinson phase would destroy true long-range order.

minor comments (6)
  1. [Eq. (I.24) vs. Eq. (IX.25)] The large-|S| asymptotic prefactor of F_h(S) is written as √(πα) in Eq. (I.24) but as α√π in Eq. (IX.25). The steepest-descent calculation in Appendix D yields α√π, so the introduction should be corrected for consistency.
  2. [Eq. (I.26)] The stated proportionality A_φ ∝ b^{-16/5} exp(-b^{-19/5} × O(1)) omits the factor b^{8χ/5} coming from L_NL^{-2χ} once L_NL ∝ b^{-4/5} exp(C b^{-19/5}) is inserted. With χ ≈ 0.388 the correct power prefactor is b^{(-16+8χ)/5} ≈ b^{-2.58}. The subsequent ratio ξ_v/L_NL ∝ b^{8/(5χ)} is unaffected, but the displayed prefactor should be corrected.
  3. [Eq. (VII.12)] The numerical prefactor 7.13 appears inconsistent with the expression in Eq. (VII.11) for z_a = 5/4. The cycle average ⟨|sin bt|^{z_a}⟩ evaluates to about 0.56 for z_a = 5/4, so the prefactor in Eq. (VII.12) should be correspondingly ≈ 0.56 unless an additional factor of four is intended; please check this numerical constant.
  4. [Section VII, Eqs. (VII.6)–(VII.15)] The cycle-averaging of the instantaneous achiral damping rate is an approximation whose validity is marginal precisely at the crossover, where the mode lifetime and the rotation period are comparable by construction. Because the quantitative scales L_c, τ_c, L_NL, and τ_NL depend on this step, the authors should state explicitly that this is an assumption rather than a derived result.
  5. [Abstract] The sentence “the hydrodynamics of a system with reasonable size is expected to governed by the linear regime” should read “is expected to be governed by the linear regime.”
  6. [Fig. 2 caption] The caption would be clearer if it identified the horizontal and vertical axes and defined the symbols L_c, L_NL, and L_v directly in the caption, since the text refers to these scales before they are introduced formally.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the KPZ mapping is derived in-paper from a stated hydrodynamic model, with external simulations supplying the only nontrivial exponents.

full rationale

The paper's central claim, that fluctuations of a 2D chiral dry Malthusian flock about the rotating 'time cholesteric' are governed by the (2+1)-dimensional KPZ equation, is derived in the paper rather than assumed: Eq. (IV.6) is obtained by inserting the rotating-phase ansatz (III.2) into the stated chiral hydrodynamic model (II.4), projecting orthogonal to the velocity, and time-averaging over one rotation cycle using (IV.5); Appendix A then extends the argument to all symmetry-allowed terms and shows they reduce to the same two terms. The mapping is not defined in terms of the target result, and no fitted parameter is relabeled as a prediction. The b-dependence of nu and lambda_K (Eqs. VII.18 and VII.28) follows from the achiral dynamical exponent z_a taken from independent simulations by Chate and Solon [38], with the authors explicitly using only the simulated exponents and not [38]'s analytic argument; the KPZ roughness and dynamical exponents chi and z are likewise taken from external numerics [27-32]. The crossover scales L_c, L_NL, tau_c, and tau_NL are derived from matching achiral damping to the rotation frequency and from the standard KPZ renormalization group, not from fits to the quantities being predicted. The vortex section is explicitly labeled speculative but is peripheral to the abstract's short-range-order claim, which already follows from the divergent fluctuations of a massless two-dimensional phase field. The reliance on the authors' earlier achiral Malthusian theory [15-17] is a standard hydrodynamic-model input that is externally benchmarked by [38], and hence is not a circular reduction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles or fields; it reuses the time-cholesteric state and vortices from prior literature. The central derivation relies on generic hydrodynamic coefficients, external numerical exponents, and a heuristic cycle-averaging step that is not formally controlled.

free parameters (4)
  • chirality frequency b
    External control parameter; the paper derives scaling laws in b but does not determine b from the theory.
  • non-universal constants C_alpha, C_NL, C_vL
    Amplitudes entering L_NL, tau_NL, L_v, and tau_v; left as undetermined material-dependent constants.
  • density coupling coefficients e_3 and e_4
    Amplitudes of the enslaved density response; not needed for exponents but needed for absolute density correlations.
  • phase noise strength D_phi
    Assumed independent of b in the small-chirality limit; sets the overall scale of phase fluctuations.
assumptions (5)
  • domain assumption Achiral Malthusian hydrodynamic equations from refs [15-17] are the correct starting point.
    Used at the outset in Section II to build the chiral velocity EOM and to define the achiral dynamic exponents.
  • domain assumption Fluctuations of |v| are massive and can be frozen.
    Footnote [49]; allows eliminating velocity magnitude fluctuations and describing order by the phase alone.
  • ad hoc to paper Time-averaging over one rotation cycle captures the slow phase dynamics.
    Section IV; needed to derive the KPZ equation from the chiral velocity EOM. The separation of fast rotation from slow phase is assumed, not rigorously controlled.
  • domain assumption Achiral exponents z_a and zeta_a from simulations [38] and KPZ exponents chi and z from [27-32] are correct.
    Used for numerical values in Table I and for all quantitative predictions of the hierarchy.
  • ad hoc to paper Vortex behavior of compact KPZ follows the 2D XY analogy.
    Section XI; the authors explicitly call this speculative, but they use it to predict L_v and tau_v.

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Cite this review

Pith. "Pith review of Hydrodynamic Theory of Two-dimensional Chiral Malthusian Flocks." pith.science (2026). https://pith.science/paper/YW5EDT4K

@misc{pith2026250717762,
  author       = {Pith},
  title        = {Pith review of: Hydrodynamic Theory of Two-dimensional Chiral Malthusian Flocks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YW5EDT4K}},
  note         = {Machine review of arXiv:2507.17762}
}
read the original abstract

We study the hydrodynamic behavior of two-dimensional chiral dry Malthusian flocks; that is, chiral polar-ordered active matter with neither number nor momentum conservation. We show that, in the absence of fluctuations, such systems generically form a ``time cholesteric", in which the velocity of the entire system rotates uniformly at a fixed frequency b. Fluctuations about this state belong to the universality class of (2+1)-Kardar-Parisi-Zhang (KPZ) equation, which implies short-ranged orientational order in the hydrodynamic limit. We then show that, in the limit of weak chirality, the hydrodynamics of a system with reasonable size is expected to governed by the linear regime of the KPZ equation, exhibiting quasi-long-ranged orientational order. Our predictions for the velocity and number density correlations are testable in both simulations and experiments.

Figures

Figures reproduced from arXiv: 2507.17762 by the authors.

Figure 1
Figure 1. FIG. 1. (a) In a generic 2D chiral Malthusian flock, the mean [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Regimes of different behavior in the limit of weak [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Plot of the temporally Fourier transformed corre [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 3
Figure 3. Figure 3: Note that the oscillating and exponentially de [PITH_FULL_IMAGE:figures/full_fig_p021_3.png]
Figure 4
Figure 4. Figure 4: FIG. 4. The illustration of the closed contour of the contour [PITH_FULL_IMAGE:figures/full_fig_p032_4.png]

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Reference graph

Works this paper leans on

73 extracted references · 72 canonical work pages

  1. [55]

    Dynamic Scaling of Two- Dimensional Polar Flocks

    H. Chat´ e and A. Solon, “Dynamic Scaling of Two- Dimensional Polar Flocks”, Phys. Rev. Lett.132, 268302 (2024). We believe the analytic argument presented in this paper that ostensibly obtains exact values for the 38 exponentsz a,ζ a, andχ a (which are calledz,ζ, andχ in that paper) is incorrect, as argued in [39]. Here we use only the exponents Chate an...

    work page 2024

  2. [38]

    Condensa- tion and Synchronization in Aligning Chiral Active Mat- ter

    Y. Wang, B. Vent´ ejou, H. Chat´ e, X. Q. Shi, “Condensa- tion and Synchronization in Aligning Chiral Active Mat- ter”, Phys. Rev. Lett.133, 258302 (2024)

    work page 2024

  3. [1]

    left over“, with all of the others contracted with others on the left, or three velocities “left over“. We’ll now consider these cases in turn. One velocity “left over

    One-derivative terms a. achiral one derivative terms The most generalachiralone-derivative term in the EOM (that is, the expression for∂ tvi) can be written[56] as λn αTijkl...mnp vjvk.vl...∂nvp ,(A.1) where the 2nindex tensorT(withnan integer) is composed of the product ofnKronecker delta‘s, which contract all but one of the indices in the product vjvkvl...

    work page

  4. [2]

    the remaining velocity component on the left is con- tracted with thenof the∂ nvp,

    work page

  5. [3]

    the remaining velocity component on the left is con- tracted with thepof the∂ nvp,

    work page

  6. [4]

    free” indexi. In case 1), the remaining index “p

    the remaining velocity component on the left is the “free” indexi. In case 1), the remaining index “p” must bei, and (A.1) reduces to λn αv2n−2 0 vj∂jvi =λ n αnv2n−2 0 (v· ∇v)i .(A.2) Sincev 0 is a constant, the sum of all such possible terms is simply λ1(v· ∇v)i ; (A.3) withλ 1 = P∞ n=1 λn αvn 0 2n−2. This is exactly theλ 1 term we had in our EOM (II.4)....

    work page

  7. [5]

    Hence all of these terms vanish as well

    = 0,(A.6) wherewis some index, and it doesn’t matter which. Hence all of these terms vanish as well. The only other possibility is that one of the three ve- locities to the left of the derivative contracts with the derivative, and the other two contract with each other. It is easy to see that this simply leads to another contri- bution to theλ 1 term equa...

    work page

  8. [6]

    terms with two derivatives a. achiral two derivative terms There are two types of such terms: λ(2.1) α T (2.1) ijkl...mnps vnvpvs...(∂jvk)(∂lvm),(A.8) and λ(2.2) α T (2.2) ijkl...mnps vnvpvs...(∂j∂lvm),(A.9) where, as before the tensorsT (2.1) andT (2.2) are prod- ucts of Kronecker deltas that contract the indices of the velocities and partial derivatives...

    work page

Show all 73 references
  1. [7]

    In this case, when we “project” this term orthogonal to vby acting on it withϵ isvs, it will vanish

    two of the outside velocities contract with the deriva- tives, and the remaining one is the free indexi(remember that we are looking at a term in the expression for∂ tvi). In this case, when we “project” this term orthogonal to vby acting on it withϵ isvs, it will vanish

  2. [8]

    This leaves three possibilities: vj(∂jvi)(∂lvl),(1.2.1),(A.11) vk(∂kvj)(∂jvi),(1.2.2),(A.12) and vk(∂kvj)(∂ivj),(1.2.3).(A.13) Consider first (1.2.1)

    two of the outside indices contract with each other, and the remaining index contracts with one of the derivatives. This leaves three possibilities: vj(∂jvi)(∂lvl),(1.2.1),(A.11) vk(∂kvj)(∂jvi),(1.2.2),(A.12) and vk(∂kvj)(∂ivj),(1.2.3).(A.13) Consider first (1.2.1). Using ∂jvi...

  3. [9]

    This gives us a termv 2n 0 ∂j∂jvi, which is proportional to the µ1∇2vi term we had in our original EOM (II.4)

    All velocities outside the derivatives contract with each other, and the two derivatives contract with each other, leaving the differentiated velocity to have indexi. This gives us a termv 2n 0 ∂j∂jvi, which is proportional to the µ1∇2vi term we had in our original EOM (II.4)

  4. [10]

    This gives us a term proportional to theµ 2∂i∇ ·vterm we had in our original EOM (II.4)

    All velocities outside the derivatives contract with each other, and one of the derivative indices contracts with the differentiated velocity, and the other isi. This gives us a term proportional to theµ 2∂i∇ ·vterm we had in our original EOM (II.4)

  5. [11]

    This gives us theµ 3(v· ∇)2vi term in our original EOM (II.4)

    All undifferentiated velocities but two contract with each other, and the two remaining undifferentiated veloc- ities contract with the derivatives, leaving the differenti- ated velocity index to bei. This gives us theµ 3(v· ∇)2vi term in our original EOM (II.4)

  6. [12]

    All undifferentiated velocities but two contract with each other, and one of the two remaining undifferenti- ated velocities contracts with the differentiated velocity, and the other contracts with one of the derivatives, leav- ing the remaining derivative index to bei. This g...

  7. [13]

    27 So our original EOM (II.4) contained all possible achi- ral two derivative terms

    = 0. 27 So our original EOM (II.4) contained all possible achi- ral two derivative terms. b. chiral two derivative terms As for the one-derivative terms, chiral two-derivative terms can be reduced to terms involving only oneϵma- trix since the product of two of them annihilate...

  8. [14]

    concentration

    Whenλ K →0withbremaining finite In this case, the large value ofbmakes the crossover to chiral behavior happen almost immediately; that is, at small length and time scales. Hence, there is no sub- stantial renormalization of the diffusion constants in the achiral regime, becau...

  9. [15]

    Whenb→0withλ K remaining finite In this case, we still get the fluctuation induced en- hancement of the diffusivityν∝b −ην given by equation (VII.28). But now, because the non-linearityλ K remains finite asb→0, the scaling law relating the bare valueg 0 ofgtobnow changes to g0...

  10. [16]

    Hence, thatn= 0 maximum (D.25) is the absolute maximum of Re h Ei − 1 vei π 4 i over all positivev

    maximum given by (D.25). Hence, thatn= 0 maximum (D.25) is the absolute maximum of Re h Ei − 1 vei π 4 i over all positivev. This leads immediately to the bound (D.19). It follows that Re g(ve i π 4 ) = Re i|S|ve i π 4 + α 2 Re Ei − 1 vei π 4 <Re i|S|ve i π 4 + αC 2 <− |S|v√ 2...

  11. [17]

    That term is 1 6 d3 Re Φ vei π 4 dv3 v= 1√ |S| ∆3 3 = |S|2 √ 2 ∆3 3 .(D.54) To ensure that this is≪1 asS→ ∞, we must have ∆3 ≪ |S|− 2 3 ,(D.55) which is consistent with (D.46)

    term in the expansion is≪1. That term is 1 6 d3 Re Φ vei π 4 dv3 v= 1√ |S| ∆3 3 = |S|2 √ 2 ∆3 3 .(D.54) To ensure that this is≪1 asS→ ∞, we must have ∆3 ≪ |S|− 2 3 ,(D.55) which is consistent with (D.46). Now using (D.53) in the bound (D.52), and using that bound in the bound ...

  12. [18]

    Toner,The Physics of Flocking: Birth, Death, and Flight in Active Matter, Cambridge University Press, (2024)

    J. Toner,The Physics of Flocking: Birth, Death, and Flight in Active Matter, Cambridge University Press, (2024)

    2024

  13. [19]

    The Mechanics and Statistics of Ac- tive Matter

    S. Ramaswamy, “The Mechanics and Statistics of Ac- tive Matter”, Ann. Rev. Condens. Matt. Phys.1, 323-345 (2010)

    2010

  14. [20]

    Hydrody- namics of soft active matter

    M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T.B. Liv- erpool, J. Prost, M. Rao, and R. A. Simha, “Hydrody- namics of soft active matter”, Rev. Mod. Phys.85, 1143- 1188 (2013)

    2013

  15. [21]

    Novel Type of Phase Transition in a System of Self-Driven Particles

    T. Vicsek, A. Czir´ ok, E. Ben-jacob, I. Cohen, and O. Shochet, “Novel Type of Phase Transition in a System of Self-Driven Particles”, Phys. Rev. Lett.75, 1226 (1995)

    1995

  16. [22]

    Long-Range Order in a Two- Dimensional DynamicalXYModel: How Birds Fly To- gether

    J. Toner, and Y. Tu, “Long-Range Order in a Two- Dimensional DynamicalXYModel: How Birds Fly To- gether”, Phys. Rev. Lett.75, 4326 (1995)

    1995

  17. [23]

    Flocks, herds, and schools: A quan- titative theory of flocking

    J. Toner, and Y. Tu, “Flocks, herds, and schools: A quan- titative theory of flocking”, Phys. Rev. E58, 4828(1998)

    1998

  18. [24]

    Hydrodynamics and phases of flocks

    J. Toner, Y. Tu, and S. Ramaswamy, “Hydrodynamics and phases of flocks”, Ann. Phys.318, 170 (2005)

    2005

  19. [25]

    Active particles in complex and crowded environments

    C. Bechinger, R. Di Leonardo, H. L¨ owen, C. Reichhardt, G. Volpe, and G. Volpe, “Active particles in complex and crowded environments”, Rev. Mod. Phys.88, 045006 (2016)

    2016

  20. [26]

    Schweitzer,Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences

    F. Schweitzer,Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences. Springer Series in Synergetics (Springer, New York, 2003). 37

    2003

  21. [27]

    Fluidization of Tissues by Cell Division and Apoptosis

    J. Ranft, M. Basan, J. Elgeti, J.-F. Joanny, J. Prost and F. J¨ ulicher, “Fluidization of Tissues by Cell Division and Apoptosis”, Proc. Natl. Acad. Sci. USA107, 20863 (2010); J. Ranft, J. Prost, F. J¨ ulicher, and J.-F. Joanny, “Tissue dynamics with permeation”, Eur. Phys. J....

    2010

  22. [28]

    Arrested phase separation in reproducing bac- teria creates a generic route to pattern formation

    M. E. Cates, D. Marenduzzo, I. Pagonabarraga, and J. Tailleur, “Arrested phase separation in reproducing bac- teria creates a generic route to pattern formation”, Proc. Natl. Acad. Sci. USA107(26), 11715 (2010)

    2010

  23. [29]

    A growing bacterial colony in two dimensions as an active nematic

    D. Dell’Arciprete, M. L. Blow, A. T. Brown, F. D. C. Farrell, J. S. Lintuvuori, A. F. McVey, D. Marenduzzo and W. C. K. Poon, “A growing bacterial colony in two dimensions as an active nematic”, Nature Communica- tions9, 4190 (2018); Z. You, D. J. G. Pearce, A. Sen- gupta, and...

    2018

  24. [30]

    Onset of Collective and Co- hesive Motion

    G. Gr´ egoire and H. Chat´ e, “Onset of Collective and Co- hesive Motion”, Phys. Rev. Lett.92, 025702 (2004)

    2004

  25. [32]

    Birth, Death, and Flight: A Theory of Malthu- sian Flocks

    J. Toner, “Birth, Death, and Flight: A Theory of Malthu- sian Flocks”, Phys. Rev. Lett.108, 088102 (2012)

    2012

  26. [33]

    Moving, Reproduc- ing, and Dying Beyond Flatland: Malthusian Flocks in Dimensionsd >2

    L. Chen, C. F. Lee, and J. Toner, “Moving, Reproduc- ing, and Dying Beyond Flatland: Malthusian Flocks in Dimensionsd >2”, Phys. Rev. Lett.125, 098003 (2020)

    2020

  27. [34]

    Universality class for a nonequilibrium state of matter: Ad= 4−ϵexpansion study of Malthusian flocks

    L. Chen, C. F. Lee, and J. Toner, “Universality class for a nonequilibrium state of matter: Ad= 4−ϵexpansion study of Malthusian flocks”, Phys. Rev. E102, 022610 (2020)

    2020

  28. [35]

    Hydrodynamic Fluc- tuations and Instabilities in Ordered Suspensions of Self- Propelled Particles

    S. Ramaswamy and R. A. Simha, “Hydrodynamic Fluc- tuations and Instabilities in Ordered Suspensions of Self- Propelled Particles”, Phys. Rev. Lett.89, 058101 (2002); “Statistical hydrodynamics of ordered suspensions of self- propelled particles: waves, giant number fluctuation...

    2002

  29. [36]

    Rheology of Active-Particle Suspensions

    Y. Hatwalne, S. Ramaswamy, M. Rao and R. A. Simha, “Rheology of Active-Particle Suspensions”, Phys. Rev. Lett.92118101 (2004)

    2004

  30. [37]

    Weak Chirality in Ordered Dna Phases

    R. D. Kamien, “Weak Chirality in Ordered Dna Phases”, Mol. Cryst. Liq. Cryst.299, 265 (1997); A. B. Harris, R. D. Kamien, and T. C. Lubensky, “Microscopic Origin of Cholesteric Pitch”, Phys. Rev. Lett.78, 1476 (1997); er- ratum Phys. Rev. Lett.78, 2867 (1997); A. B. Harris, R....

    1997

  31. [39]

    Chiral active matter

    B. Liebchen, and D. Levis, “Chiral active matter”, EPL, 139, 67001 (2022)

    2022

  32. [40]

    Collective Behavior of Chiral Active Matter: Pattern Formation and Enhanced Flock- ing

    B. Liebchen, and D. Levis, “Collective Behavior of Chiral Active Matter: Pattern Formation and Enhanced Flock- ing”, Phys. Rev. Lett.119, 058002 (2017)

    2017

  33. [41]

    left-handed

    However, it should be noted that, in general, there is no unique measure of the chirality of a chiral system. Hence, it is therefore possible that, by tuning one parameter of a chiral flock (e.g., by mixing “left-handed” particles into a “right-handed” flock), one might tune t...

  34. [42]

    Associated short paper

  35. [43]

    Dynamic scaling of growing interfaces

    M. Kardar, G. Parisi and Y.-C. Zhang, “Dynamic scaling of growing interfaces”, Physical Review Letters56, 889- 892 (1986)

    1986

  36. [44]

    Surface Roughening in a Hypercube-Stacking Model

    B. Forrest and L. H. Tang, “Surface Roughening in a Hypercube-Stacking Model”, Phys. Rev. Lett.64, 1405 (1990)

    1990

  37. [45]

    Extremely large-scale simu- lation of a Kardar-Parisi-Zhang model using graphics cards

    J. Kelling and G. `Odor, “Extremely large-scale simu- lation of a Kardar-Parisi-Zhang model using graphics cards”, Phys.Rev. E84, 061150 (2011)

    2011

  38. [46]

    Dynamical uni- versality classes of simple growth and lattice gas models

    J. Kelling, G. `Odor, and S. Gemming, “Dynamical uni- versality classes of simple growth and lattice gas models”, J. Phys. A: Math. Theor.51, 035003 (2018)

    2018

  39. [47]

    (2+1)-Dimensional Directed Polymer in a Random Medium: Scaling Phenomena and Universal Distributions

    T. Halpin-Healy, “(2+1)-Dimensional Directed Polymer in a Random Medium: Scaling Phenomena and Universal Distributions”, Phys. Rev. Lett.109, 170602 (2012)

    2012

  40. [48]

    Kardar-Parisi-Zhang growth inϵdi- mensions and beyond

    T. Halpin-Healy, “Kardar-Parisi-Zhang growth inϵdi- mensions and beyond”, preprint (2024)

    2024

  41. [49]

    Numerical estimate of the Kardar-Parisi-Zhang universality class in (2+1) dimen- sions

    A. Pagnani and G. Parisi, “Numerical estimate of the Kardar-Parisi-Zhang universality class in (2+1) dimen- sions”, Phys. Rev. E92, 010101(R) (2015)

    2015

  42. [50]

    Temperature effect on (2 + 1) experimental Kardar-Parisi-Zhang growth

    R. A. L. Almeida, S. O. Ferreira, I. R. B. Ribeiro and T. J. Oliveira, “Temperature effect on (2 + 1) experimental Kardar-Parisi-Zhang growth”, EPL109,46003 (2015)

    2015

  43. [51]

    Surprising mappings of 2D polar active fluids to 2D soap and 1D sandblasting

    L. Chen, C.F. Lee and J. Toner, “Surprising mappings of 2D polar active fluids to 2D soap and 1D sandblasting”, Nat. Commun.7, 12215 (2016)

    2016

  44. [52]

    Emergent Kardar-Parisi-Zhang Phase in Quadratically Driven Con- densates

    O. Diessel, S. Diehl, and A. Chiocchetta, “Emergent Kardar-Parisi-Zhang Phase in Quadratically Driven Con- densates”, Phys. Rev. Lett.128, 070401 (2022)

    2022

  45. [53]

    Emergent po- lar order in nonpolar mixtures with nonreciprocal inter- actions

    G. Pisegna, S. Saha, and R. Golestanian, “Emergent po- lar order in nonpolar mixtures with nonreciprocal inter- actions”, PNAS121, e2407705121 (2024)

    2024

  46. [54]

    Kardar-Parisi-Zhang scaling in time-crystalline mat- ter

    R. Daviet, C. P. Zelle, A. Asadollahi, and S. Diehl, “Kardar-Parisi-Zhang scaling in time-crystalline mat- ter”, arXiv:2412.09677

    arXiv

  47. [56]

    The inconvenient truth about flocks

    L. Chen, P. Jentsch, C. F. Lee, A. Maitra, S. Ra- maswamy, and J. Toner, “The inconvenient truth about flocks”, arXiv:2503.17064

    arXiv

  48. [57]

    Ordering, metasta- bility and phase transitions in two-dimensional systems

    J. M. Kosterlitz and D. J. Thouless, “Ordering, metasta- bility and phase transitions in two-dimensional systems”, J. Phys. C: Solid State Phys.6, 1181 (1973)

    1973

  49. [58]

    Two-Dimensional Superfluidity of Exciton Polaritons Requires Strong Anisotropy

    See, e.g., E. Altman, L. M. Sieberer, L. Chen, S. Diehl, and J. Toner, “Two-Dimensional Superfluidity of Exciton Polaritons Requires Strong Anisotropy”, Phys. Rev. X5, 011017 (2015)

    2015

  50. [59]

    Live Soap: Stability, Order, and Fluctuations in Apolar Ac- tive Smectics

    T. C. Adhyapak, S. Ramaswamy, and J. Toner, “Live Soap: Stability, Order, and Fluctuations in Apolar Ac- tive Smectics”, Phys. Rev. Lett.110, 118102 (2013)

    2013

  51. [60]

    Broken living layers: Dislocations in active smectic liquid crystals

    F. J¨ ulicher, J. Prost, and J. Toner, “Broken living layers: Dislocations in active smectic liquid crystals”, Phys. Rev. E106, 054607 (2022)

    2022

  52. [61]

    The surface statistics of a granular aggregate

    S.F. Edwards and D.R. Wilkinson, “The surface statistics of a granular aggregate”, Proc. R. Soc. Lond. A38117 (1982)

    1982

  53. [62]

    De Progressionibus harmonicus observa- tiones

    L. Euler, “De Progressionibus harmonicus observa- tiones”, paper presented to the St. Petersburg Academy, (1734)

  54. [63]

    I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series, and Products“, Academic press, New York (1980); eqn. (8.214), pg. 927

    1980

  55. [64]

    Col- lective motion of self-propelled particles interacting with- out cohesion

    H. Chat´ e, F. Ginelli, G. Gr´ egoire, and F. Raynaud, “Col- lective motion of self-propelled particles interacting with- out cohesion”, Phys. Rev. E77, 046113 (2008)

    2008

  56. [65]

    Active nematics on a substrate: Giant number fluctuations and long-time tails

    S. Ramaswamy, R. Aditi Simha and J. Toner, “Active nematics on a substrate: Giant number fluctuations and long-time tails”, EPL62, 196 (2003)

    2003

  57. [66]

    soft-spin

    Our approach here of freezing the magnitude ofvand only allowing its orientation to fluctuate differs from the approach of [15–17], who used a “soft-spin” model in which the magnitude|v|of the velocity fieldvwas al- lowed to fluctuate as well. However, in those papers, and in ...

  58. [67]

    Minimum Scaling Model and Exact Exponents for the Nambu-Goldstone Modes in the Vicsek Model

    H. Ikeda, “Minimum Scaling Model and Exact Exponents for the Nambu-Goldstone Modes in the Vicsek Model”, Phys. Rev. Lett.133, 258301 (2024)

    2024

  59. [68]

    P. G. de Gennes and J. Prost,The Physics of Liquid Crystals, 2nd ed. (Clarendon Press, Oxford, 1993)

    1993

  60. [69]

    Caille, ”Remarques sur la diffusion des rayons X dans les smectiques”, C.R

    A. Caille, ”Remarques sur la diffusion des rayons X dans les smectiques”, C.R. Acad. Sci. Ser. 8274, 891 (1972); T. C. Lubensky, ”Low Temperature phase of infinite cholesterics”, Phys. Rev. Lett. 29, 206 (1972); P. G. De- Gennes, ”Conjectures sur l’´ etat smectique” , J. Phys....

    1972

  61. [70]

    A difference between smectics and our system is that here we donothave any peaks at higher harmonicsω=mb with integer|m|>1. This is because the temporal mod- ulation in our system is, in the absence of fluctuations, a perfect sine wave, while in smectics, the spatial mod- ulat...

  62. [71]

    Non-reciprocal phase transitions

    M. Fruchart, R. Hanai, P. B. Littlewood, V. Vitelli, “Non-reciprocal phase transitions”, Nature592, 363 (2021)

    2021

  63. [72]

    Activity unmasks chirality in liquid- crystalline active matter

    A. Maitra, “Activity unmasks chirality in liquid- crystalline active matter”, Ann. Rev. Cond. Mat. Phys. 16, 275 (2025)

    2025

  64. [73]

    Terms in which the derivative acts on more than one velocity component - i.e., terms in which there is more than one velocity component to the right of the deriva- tive operator- can be reduced by the product rule for derivatives to terms of the form (A.1)

  65. [74]

    However, because the integrand here is analytic, we can deform the con- tour of integration into that illustrated in figure 4, with θ0 =π/4

    The alert reader (thepreternaturallyalert reader) will no- tice that this change of variables makes the upper limit of the integral ei π 4 ∞, rather than real∞. However, because the integrand here is analytic, we can deform the con- tour of integration into that illustrated in...

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