We present a quantitative assessment of the Toner and Tu theory describing the universal scaling of fluctuations in polar phases of dry active matter. Using large scale simulations of the Vicsek model in two and three dimensions, we find the overall phenomenology and generic algebraic scaling predicted by Toner and Tu, but our data on density correlations reveal some qualitative discrepancies. The values of the associated scaling exponents we estimate differ significantly from those conjectured in 1995. In particular, we identify a large crossover scale beyond which flocks are only weakly anisotropic. We discuss the meaning and consequences of these results.
We study clustering and percolation phenomena in the Vicsek model, taken here in its capacity of prototypical model for dry aligning active matter. Our results show that the order-disorder transition is not related in any way to a percolation transition, contrary to some earlier claims. We study geometric percolation in each of the phases at play, but we mostly focus on the ordered Toner-Tu phase, where we find that the long-range correlations of density fluctuations give rise to a novel anisotropic percolation transition.
We study numerically the Toner-Tu field theory where the density field is maintained constant, a limit case of "Malthusian" flocks for which the asymptotic scaling of correlation functions in the ordered phase is known exactly. While we confirm these scaling laws, we also show that such constant-density flocks are metastable to the nucleation of a specific defect configuration, and are replaced by a globally disordered phase consisting of asters surrounded by shock lines that constantly evolves and remodels itself. We demonstrate that the main source of disorder lies along shock lines, renderi…
How Continuous Symmetry Stabilizes the Ordered Phase of Polar Flocks
University of Chicago（芝加哥大学）；CEA, CNRS Université Paris-Saclay（巴黎萨克雷大学CEA/CNRS）；Computational Science Research Center, Beijing（北京计算科学研究中心）；Sorbonne Université, CNRS（索邦大学/CNRS）；Technion – Israel Institute of Technology（以色列理工学院）；Harvard University（哈佛大学）；Massachusetts Institute of Technology（麻省理工学院）
A recent kinetic approach for Vicsek-like models of active particles is reviewed. The theory is based on an exact Chapman-Kolmogorov equation in phase space. It can handle discrete time dynamics and "exotic" multi-particle interactions. A nonlocal mean-field theory for the one-particle distribution function is obtained by assuming molecular chaos. The Boltzmann approach of Bertin et al., Phys. Rev. E 74, 022101 (2006) and J. Phys. A: Math. Theor. 42, 445001 (2009), is critically assessed and compared to the current approach. In Boltzmann theory, a collision starts when two particles enter each…
A collection of self-propelled particles (SPPs) shows coherent motion and exhibits a true long range ordered (LRO) state in two dimensions. Various studies show that the presence of spatial inhomogeneities can destroy the usual long-range ordering in the system. However, the effects of inhomogeneity due to the intrinsic properties of the particles are barely addressed. In this paper we consider a collection of polar SPPs moving with inhomogeneous speed (IS) on a two dimensional substrate, which can arise due to varying physical strength of the individual particle. To our surprise, the IS not o…
We show that spatial quenched disorder affects polar active matter in ways more complex and far-reaching than believed heretofore. Using simulations of the 2D Vicsek model subjected to random couplings or a disordered scattering field, we find in particular that ergodicity is lost in the ordered phase, the nature of which we show to depend qualitatively on the type of quenched disorder: for random couplings, it remains long-range ordered, but qualitatively different from the pure (disorderless) case. For random scatterers, polar order varies with system size but we find strong non-self-averagi…
We study a collection of polar self-propelled particles (SPPs) on a two-dimensional substrate in the presence of random quenched rotators. These rotators act like obstacles which rotate the orientation of the SPPs by an angle determined by their intrinsic orientations. In the zero self-propulsion limit, our model reduces to the equilibrium $XY$ model with quenched disorder, while for the clean system, it is similar to the Vicsek model for polar flock. We note that a small amount of the quenched rotators destroys the long-range order usually noted in the clean SPPs. The system shows a quasi-lon…
We study the polar collective dynamics of Janus colloidal particles fueled by an AC electric field. When the density is high enough, the polar interactions between the particles induce a polar orientationally ordered state which exhibits features reminiscent of the Vicsek model such as true long-range order and giant number fluctuations. Independent measurements of the polarity and velocity at the single particle level allowed us to investigate the single particle dynamics within the ordered state. We discovered theoretically-unaddressed statistical properties of the ordered state such as the …
In this study, we introduce a minimal model for a collection of polar self-propelled particles (SPPs) on a two-dimensional substrate where each particle has a different ability to interact with its neighbours. The SPPs interact through a short-range alignment interaction and interaction strength of each particle is obtained from a uniform distribution. Moreover, the volume exclusion among the SPPs is taken care of by introducing a repulsive interaction among them. We characterise the ordered steady state and kinetics of the system for different strengths of the disorder. We find that the prese…
As the constituent particles of a flock are polar and in a driven state, their interactions must, in general, be fore-aft asymmetric and non-reciprocal. Within a model that explicitly retains the classical spin angular momentum field of the particles we show that the resulting asymmetric contribution to interparticle torques, if large enough, leads to a buckling instability of the flock. Precisely this asymmetry also yields a natural mechanism for a difference between the speed of advection of information along the flock and the speed of the flock itself, concretely establishing that the absen…
We consider the off-lattice two-dimensional $q$-state active clock model (ACM) as a natural discretization of the Vicsek model (VM) describing flocking. The ACM consists of particles able to move in the plane in a discrete set of $q$ equidistant angular directions, as in the active Potts model (APM), with an alignment interaction inspired by the ferromagnetic equilibrium clock model. We find that for a small number of directions, the flocking transition of the ACM has the same phenomenology as the APM, including macrophase separation and reorientation transition. For a larger number of directi…
I reanalyze the hydrodynamic theory of fluid, polar ordered flocks. I find new linear terms in the hydrodynamic equations which slightly modify the anisotropy, but not the scaling, of the damping of sound modes. I also find that the nonlinearities allowed {\it in equilibrium} do not stabilize long ranged order in spatial dimensions $d=2$; in accord with the Mermin-Wagner theorem. Nonequilibrium nonlinearities {\it do} stabilize long ranged order in $d=2$, as argued by earlier work. Some of these were missed by earlier work; it is unclear whether or not they change the scaling exponents in $d=2…
Elongated active units cannot spontaneously break rotation symmetry in bulk fluids to form nematic or polar phases. This has led to the image of active suspensions as spontaneously evolving, spatiotemporally chaotic fluids. In contrast, I show that bulk active fluids have stable active nematic and polar states at fluid-fluid or fluid-air interfaces. The active flow-mediated long-range interactions that destroy the ordered phase in bulk, lead to long-range order at the interface. The active fluids have a surface ordering transition and form states with quiescent, ordered surfaces and a chaotic …
We studied a system of polar self-propelled particles (SPPs) on a thin rectangular channel designed into three regions of order-disorder-order. The division of the three regions is made on the basis of the noise SPPs experience in the respective regions. The noise in the two wide region is chosen lower than the critical noise of order-disorder transition and noise in the middle region or interface is higher than the critical noise. This make the geometry of the system analogous to the Josephson Junction (JJ) in solid state physics. Keeping all other parameters fixed, we study the properties of…
We study the asymptotic response of polar ordered active fluids ("flocks") to small external aligning fields $h$. The longitudinal susceptibility $χ_{_\parallel}$ diverges, in the thermodynamic limit, like $h^{-ν}$ as $h \rightarrow 0$. In finite systems of linear size $L$, $χ_{_\parallel}$ saturates to a value $\sim L^γ$. The universal exponents $ν$ and $γ$ depend only on the spatial dimensionality $d$, and are related to the dynamical exponent $z$ and the "roughness exponent" $α$ characterizing the unperturbed flock dynamics. Using a well supported conjecture for the values of these two expo…
We present a comprehensive theory of the dynamics and fluctuations of a two-dimensional suspension of polar active particles in an incompressible fluid confined to a substrate. We show that, depending on the sign of a single parameter, a state with polar orientational order is anomalously stable (or anomalously unstable), with a nonzero relaxation (or growth) rate for angular fluctuations at zero wavenumber. This screening of the broken-symmetry mode in the stable state does lead to conventional rather than giant number fluctuations as argued by Bricard et al., Nature ${\bf 503}$, 95 (2013), b…
We study the order-disorder transition in two-dimensional incompressible systems of motile particles with alignment interactions through extensive numerical simulations of the incompressible Toner-Tu (ITT) field theory and a detailed finite-size scaling (FSS) analysis. The transition looks continuous in the explored parameter space, but the effective susceptibility exponent $γ/ν$ and the dynamic exponent $z$ exhibit a strong, non-monotonic variation on the system size in the form of double crossovers. At small sizes, mean-field exponents are observed for the homogeneous $k=0$ mode whereas spat…
We study universal behavior in the moving phase of a generic system of motile particles with alignment interactions in the incompressible limit for spatial dimensions $d>2$. Using a dynamical renormalization group analysis, we obtain the exact dynamic, roughness, and anisotropy exponents that describe the scaling behavior of such incompressible systems. This is the first time a compelling argument has been given for the exact values of the anomalous scaling exponents of a flock moving through an isotropic medium in $d>2$.
We present in detail a Langevin formalism for constructing stochastic dynamical equations for active-matter systems coupled to a thermal bath. We apply the formalism to clarify issues of principle regarding the sources and signatures of nonequilibrium behaviour in a variety of polar and apolar single-particle systems and polar flocks. We show that distance from thermal equilibrium depends on how time-reversal is implemented and hence on the reference equilibrium state. We predict characteristic forms for the frequency-resolved entropy production for an active polar particle in a harmonic poten…
We investigate the steady-state entropy production rate (EPR) in the Hydrodynamic Vicsek Model (HVM) and Diffusive Flocking Model (DFM). Both models display a transition from an isotropic gas to a polar liquid (flocking) phase, in addition to traveling polar clusters and microphase-separation in the miscibility gap. The phase diagram of the DFM, which may be considered an extension of the HVM, contains additional structure at low densities where we find a novel crystal phase in which a stationary hexagonal lattice of high-density ridges surround low density valleys. From an assessment of the s…
Starting from symmetry considerations, we derive the generic hydrodynamic equation of non-equilibrium $XY$ spin systems with $N$-atic symmetry under weak fluctuations. Through a systematic treatment we demonstrate that, in two dimensions, these systems exhibit two types of scaling behaviours. For $N=1$, they have long-range order and are described by the flocking phase of dry polar active fluids. For all other values of $N$, the systems exhibit quasi long-range order, as in the equilibrium $XY$ model at low temperature.
The hydrodynamic equations of dry active polar fluids (i.e., moving flocks without momentum conservation) are shown to imply giant number fluctuations. Specifically, the rms fluctuations $\sqrt {<(δN)^2>}$ of the number $N$ of active particles in a region containing a mean number of active particles $<N>$ scales according to the law $\sqrt {<(δN)^2>} = K'<N>^{ϕ(d)}$ with $ϕ(d)=\frac{7}{10}+\frac{1}{5d}$ in $d\le4$ spatial dimensions. This is much larger the "law of large numbers" scaling $\sqrt {<(δN)^2>} = K\sqrt{<N>}$ found in most equilibrium and non-equilibrium systems. In further contrast…
