Abstract
Ensembles of driven or motile bodies moving along opposite directions are generically reported to selforganize into strongly anisotropic lanes. Here, building on a minimal model of selfpropelled bodies targeting opposite directions, we first evidence a critical phase transition between a mingled state and a phaseseparated lane state specific to active particles. We then demonstrate that the mingled state displays algebraic structural correlations also found in driven binary mixtures. Finally, constructing a hydrodynamic theory, we single out the physical mechanisms responsible for these universal longrange correlations typical of ensembles of oppositely moving bodies.
Introduction
Should you want to mix two groups of pedestrians, or two ensembles of colloidal beads, one of the worst possible strategies would be pushing them towards each other. Both experiments and numerical simulations have demonstrated the segregation of oppositely driven Brownian particles into parallel lanes^{1,2,3,4,5}. Even the tiniest drive results in the formation of finite slender lanes which exponentially grow with the driving strength^{5}. The same qualitative phenomenology is consistently observed in pedestrian counterflows^{6,7,8,9,10}. From our daily observation of urban traffic to laboratory experiments, the emergence of counterpropagating lanes is one of the most robust phenomena in population dynamics, and has been at the very origin of the early description of pedestrians as granular materials^{11,12}. However, a description as isotropic grains is usually not sufficient to account for the dynamics of interacting motile bodies^{13,14,15}. From motilityinduced phase separation^{15}, to giant density fluctuations in flocks^{13,16,17}, to pedestrian scattering^{18,19}, the most significant collective phenomena in active matter stem from the interplay between their position and orientation degrees of freedom.
In this communication, we address the phase behaviour of a binary mixture of active particles targeting opposite directions. Building on a prototypical model of selfpropelled bodies with repulsive interactions, we numerically evidence two nonequilibrium steady states: a lane state where the two populations maximize their flux and phase separate, and a mixed state where all motile particles mingle homogeneously. We show that these two distinct states are separated by a genuine critical phase transition. In addition, we demonstrate algebraic density correlations in the homogeneous phase, akin to that recently reported for oppositely driven Brownian particles^{20}. Finally, we construct a hydrodynamic description to elucidate these longrange structural correlations, and conclude that they are universal to both active and driven ensembles of oppositely moving bodies.
Results
A minimal model of active binary mixtures
We consider an ensemble of N selfpropelled particles characterized by their instantaneous positions r_{i}(t) and orientations , where i=1, …, N (in all that follows stands for x/x). Each particle moves along its orientation vector at constant speed . We separate the particle ensemble into two groups of equal size following either the direction Θ_{i}=0 (right movers) or π (left movers) according to a harmonic angular potential . Their equations of motion take the simple form:
In principle, oriented particles can interact by both forces and torques. We here focus on the impact of orientational couplings and consider that neighbouring particles interact solely through pairwise additive torques T_{ij}. This type of model has been successfully used to describe a number of seemingly different active systems, starting from bird flocks, fish schools and bacteria colonies to synthetic active matter made of selfpropelled colloids or polymeric biofilaments^{13,21,22,23,24,25,26,27}. We here elaborate on a minimal construction where the particles interact only by repulsive torques. In practical terms, we choose the standard form , where the effective angular energy simply reads . As sketched in Fig. 1a, this interaction promotes the orientation of along the direction of the centretocentre vector r_{ij}=(r_{i}−r_{j}): as they interact particles turn their back to each other (for example, refs 24, 28, 29, 30). The spatial decay of the interactions is given by: B(r_{ij})=B(1−r_{ij}/(a_{i}+a_{j})), where B is a finite constant if r_{ij}<(a_{i}+a_{j}) and 0 otherwise. In all that follows, we focus on the regime where repulsion overcomes alignment along the preferred direction (B>1). The interaction ranges a_{i} are chosen to be polydisperse to avoid the specifics of crystallization, and we make the classic choice a=1 or 1.4 for one in every two particles. Before solving equations (1) and (2), two comments are in order. First, this model is not intended to provide a faithful description of a specific experiment. Instead, this minimal setup is used to single out the importance of repulsion torques typical of active bodies. Any more realistic description would also include hardcore interactions. However, in the limit of dilute ensembles and longrange repulsive torques, hardcore interactions are not expected to alter any of the results presented below. Second, unlike models of driven colloids or grains interacting by repulsive forces^{1,5,20}, equations (1) and (2) are not invariant upon Gallilean boosts, and therefore are not suited to describe particles moving at different speeds along the same preferred direction.
Critical mingling
Starting from random initial conditions, we numerically solve equations (1) and (2) using forward Euler integration with a time step of 10^{−2}, and a sweepandprune algorithm for neighbour summation. We use a rectangular simulation box of aspect ratio L_{x}=2L_{y} with periodic boundary conditions in both directions. We also restrain our analysis to H=1, leaving two control parameters that are the repulsion strength B and the overall density . The following results correspond to simulations with N comprised between 493 and 197,300 particles.
We observe two clearly distinct stationary states illustrated in Fig. 1b,d. At low density and/or weak repulsion the system quickly phase separates. Computing the local density difference between the right and left movers , we show that this dynamical state is characterized by a strongly bimodal density distribution, Fig. 1e. The left and right movers quickly selforganize into counterpropagating lanes separated by a sharp interface, Fig. 1b. In each stream, virtually no particle interact and most of the interactions occur at the interface, Supplementary Movie 1. As a result the particle orientations are very narrowly distributed around their mean value, Fig. 1f. In stark contrast, at high density and/or strong repulsion, the motile particles do not phase separate. Instead, the two populations mingle and continuously interact to form a homogeneous liquid phase with Gaussian density fluctuations, and much broader orientational fluctuations, Fig. 1d–f. This behaviour is summarized by the phase diagram in Fig. 1c.
Although phase separation is most often synonymous of firstorder transition in equilibrium liquids, we now argue that the lane and the mingled states are two genuine nonequilibrium phases separated by a critical line in the (B, ) plane. To do so, we first introduce the following orientational order parameter:
〈W〉 vanishes in the lane phase where on average all particles follow their preferred direction, and takes a nonzero value otherwise. We show in Fig. 2a how 〈W〉 increases with the repulsion strength B at constant . For the order parameter averages to zero below B_{c}=2.17±0.02, while above B_{c} it sharply increases as , with β=0.33±0.07, Fig. 2b. This scaling law suggests a genuine critical behaviour. We further confirm this hypothesis in Fig. 2c, showing that the fluctuations of the order parameter diverge as B−B_{c}^{−γ}, with γ=0.64±0.07. Deep in the homogeneous phase the fluctuations plateau to a constant value of the order of 1/N. Finally, the criticality hypothesis is unambiguously ascertained by Fig. 2d, which shows the powerlaw divergence of the correlation time of 〈W〉(t): with zν=1.21±0.16.
We do not have a quantitative explanation for this critical behaviour. However, we can gain some insight from the counterintuitive twobody scattering between active particles. In the overdamped limit, the collision between two passive colloids driven by an external field would at most shift their position over an interaction diameter^{31}. Here these transverse displacements are not bounded by the range of the repulsive interactions. For a finite set of impact parameters, collisions between selfpropelled particles result in persistent deviations transverse to their preferred trajectories illustrated in Fig. 3 and Supplementary Note 2. This persistent scattering stems from the competition between repulsion and alignement. When these two contributions compare, bound pairs of oppositely moving particles can even form and steadily propel along the transverse direction , Fig. 3b,c. We stress that this behaviour is not peculiar to this twobody setting: persistent transverse motion of bound pairs is clearly observed in simulations at the onset of laning, Supplementary Movie 2. We therefore strongly suspect the resulting enhanced mixing to be at the origin of the sharp melting of the lanes and the emergence of the mingled state.
Longrange correlations in mingled liquids
We now evidence longrange structural correlations in this activeliquid phase, and analytically demonstrate their universality. The overall pair correlation function of the active liquid, g(r), is plotted in Fig. 4a. At a first glance, deep in the homogeneous phase, the few visible oscillations would suggest a simple anisotropic liquid structure. However, denoting α and β the preferred direction of the populations (left or right), we find that the asymptotic behaviours of all pair correlation functions g_{αβ}(x, y=0) decay algebraically as with , Fig. 4b. This powerlaw behaviour is very close to that reported in numerical simulations^{4} and fluctuating density functional theories of oppositely driven colloids at finite temperature^{20}.
Hydrodynamic description
To explain the robustness of these longrange correlations, we provide a hydrodynamic description of the mingled state, and compute its structural response to random fluctuations. We first observe that the orientational diffusivity of the particles increases linearly with the average density in Fig. 1f inset. This behaviour indicates that binary collisions set the fluctuations of this active liquid, and hence suggests using a Boltzmann kinetictheory framework, for example, refs 32, 33 from an activematter perspective. In the large B limit, the microscopic interactions are accounted for by a simplified scattering rule anticipated from equation (2) and confirmed by the inspection of typical trajectories (Fig. 1a). Upon binary collisions the selfpropelled particles align their orientation with the centretocentre axis regardless of their initial orientation and external drive. Assuming molecular chaos and binary collisions only, the time evolution of the onepoint distribution functions ψ_{α}(r, θ, t) reads:
The convective term on the l.h.s stems from selfpropulsion, the third term accounts for alignment with the preferred direction (resp. ) for the right (resp. left) movers. Using the simplified scattering rule to express the socalled collision integral on the r.h.s., we can establish the dynamical equations for the density fluctuations δρ_{α} around the average homogeneous state (see Methods section for technical details). Within a linear response approximation, they take the compact form:
where J_{α} describes the convection and the collisioninduced diffusion of the α species, and is the coupling term, crucial to the anomalous fluctuations of the active liquid:
The two anisotropic diffusion tensors D and are diagonal and their expression is provided in Supplementary Note 3 together with all the hydrodynamic coefficients. is a particle current stemming from the fluctuations of the other species and has two origins. The first term arises from the competition between alignment along the driving direction and orientational diffusion caused by the collisions: the higher the local density , the smaller the longitudinal current. The second term originates from the pressure term ∝ ∇: a local density gradient results in a net flow of both species (see Methods section for details). This diffusive coupling is therefore generic and enters the description of any binary compressible fluid. Two additional comments are in order. First, this prediction is not specific to the smalldensity regime and is expected to be robust to the microscopic details of the interactions. As a matter of fact, the above hydrodynamic description is not only valid in the limit of strong repulsion and small densities discussed above but also in the opposite limit, where the particle density is very large while the repulsion remains finite as detailed in Supplementary Note 5. Second, the robustness of this hydrodynamic description could have been anticipated using conservation laws and symmetry considerations, as done for example, in ref. 16 for active flocks. Here the situation is simpler, momentum is not conserved and no soft mode is associated to any spontaneous symmetry breaking. As a result the only two hydrodynamic variables are the coupled (selfadvected) densities of the two populations^{34}. The associated mass currents are constructed from the only two vectors that can be formed in this homogeneous but anisotropic setting: h_{α} and ∇δρ_{α}. These simple observations are enough to set the functional form of equations (5)–(7).
By construction the above hydrodynamic description alone cannot account for any structural correlation. To go beyond this meanfield picture we classically account for fluctuations by adding a conserved noise source to equation (5) and compute the resulting densityfluctuation spectrum^{13}. At the linear response level, without loss of generality, we can restrain ourselves to the case of an isotropic additive white noise of variance 2T (Supplementary Note 4). Going to Fourier space, and after lengthy yet straightforward algebra, we obtain in the long wavelength limit:
with , and where 〈·〉 is a noise average. The crosscorrelation 〈δρ_{α}(q)δρ_{β}(−q)〉 has a similar form, Supplementary Note 4. Even though the above hydrodynamic description qualitatively differs from that of driven colloids, they both yield the same fluctuation spectra^{20}. A key observation is that the structure factor given by equation (8) is nonanalytic at q=0. Approaching q=0 from different directions yields different limits, which is readily demonstrated noting that and are both constant functions but have different values. The nonanalyticity of equation (8) in the long wavelength limit translates in an algebraic decay of the density correlations in real space. After a Fourier transform, we find: , in agreement with our numerical simulations of both selfpropelled particles, Fig. 4b, and driven colloids^{4,20}. Beyond these longrange correlations it can also be shown (Supplementary Note 4) that the pair correlation functions take the form again in excellent agreement with our numerical findings. Figure 4c,d indeed confirm that the pair correlations between both populations are correctly collapsed when normalized by x^{−3/2} and plotted versus the rescaled distance y/x^{1/2}.
Discussion
Different nonequilibrium processes can result in algebraic density correlations with different power laws, for example, ref. 35. We thus need to identify the very ingredients yielding universal decay, or equivalently structure factors of the form found both in active and driven binary mixtures. We first recall that this structure factor has been computed from hydrodynamic equations common to any system of coupled conserved fields in a homogeneous and anisotropic setting (regardless of the associated noise anisotropy,^{35} and Supplementary Note 4). The structure factor is nonanalytic as q→0, and the density correlations algebraic, only when a≠b. Inspecting equation (8), we readily see that this condition is generically fulfilled as soon as the coupling current is nonzero. In other words, as soon as the collisions between the particles either modify their transverse diffusion , or their longitudinal advection . Both ingredients are present in our model of active particles (equation (5)) and, based on symmetry considerations, should be generic to any driven binary mixtures with local interactions. Another simple physical explanation can be provided to account for the variations of the pair correlations in the transverse direction shown in Fig. 4c,d and also reported in simulations of driven particles^{20}. Selfpropulsion causes the particles to move, on average, at constant speed along the xdirection while frontal collisions induce their transverse diffusion. As a result the xposition of the particles increase linearly with time, and their transverse position increases as ∼t^{1/2}. We therefore expect the longitudinal and transverse correlations to be related by a homogeneous function of y/x^{1/2} in steady state as observed in simulations of both active and driven particles. Altogether these observations confirm the universality of the longrange structural correlations found in both classes of nonequilibrium mixtures.
In conclusion, we have demonstrated that the interplay between orientational and translational degrees of freedom, inherent to motile bodies, can result in a critical transition between a phase separated and a mingled state in binary active mixtures. In addition, we have singled out the very mechanisms responsible for longrange structural correlations in any ensemble of particles driven towards opposite directions, should they be passive colloids or selfpropelled agents.
Methods
Boltzmann kinetic theory
Let us summarize the main steps of the kinetic theory employed to establish equations (5)–(7). The socalled collision integral on the r.h.s of equation (4) includes two contributions, which translate the behaviour illustrated in Fig. 1a:
The first term indicates that a collision with any particle located at reorients the α particles along at a rate . The second term accounts for the random reorientation, at a rate , of a particle aligned with upon collision with any other particle. Within a twofluid picture, the velocity and nematic texture of the α particles are given by and . The mass conservation relation, ∂_{t}ρ_{α}+∇·(ρ_{α}V_{α})=0, is obtained by integrating equation (4) with respect to θ and constrains . The time evolution of the velocity field is also readily obtained from equation (4):
where the second term on the l.h.s is a convective term stemming from selfpropulsion. The force field on the r.h.s. of equation (10) reads: . The first term originates from the alignment of particles along the direction, the second term is a repulsioninduced pressure, and the third one echoes the collisioninduced rotational diffusivity of the particles. An additional closure relation between Q_{α}, v_{α} and ρ_{α} is required to yield a selfconsistent hydrodynamic description. Deep in the homogeneous phase, we make a wrapped Gaussian approximation for the orientational fluctuations in each population^{24,36}. This hypothesis is equivalent to setting (refs 24, 37). As momentum is not conserved, the velocity field is not a hydrodynamic variable; in the long wavelength limit the velocity modes relax much faster than the (conserved) density modes. We therefore ignore the temporal variations in equation (10) and use this simplified equation to eliminate v_{α} in the mass conservation relation, leading to the mass conservation equation (5).
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Additional information
How to cite this article: Bain, N. & Bartolo, D. Critical mingling and universal correlations in model binary active liquids. Nat. Commun. 8, 15969 doi: 10.1038/ncomms15969 (2017).
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Acknowledgements
We acknowledge support from ANR grant MiTra and Institut Universitaire de France (D.B.). We acknowledge valuable comments and suggestions by V. Démery and H. Löwen.
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D.B. designed the research. N.B. performed the numerical simulations. D.B. and N.B. performed the theory, discussed the results and wrote the paper.
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Bain, N., Bartolo, D. Critical mingling and universal correlations in model binary active liquids. Nat Commun 8, 15969 (2017). https://doi.org/10.1038/ncomms15969
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