Ensembles of driven or motile bodies moving along opposite directions are generically reported to self-organize into strongly anisotropic lanes. Here, building on a minimal model of self-propelled bodies targeting opposite directions, we first evidence a critical phase transition between a mingled state and a phase-separated 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 long-range correlations typical of ensembles of oppositely moving bodies.
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 lanes1,2,3,4,5. Even the tiniest drive results in the formation of finite slender lanes which exponentially grow with the driving strength5. The same qualitative phenomenology is consistently observed in pedestrian counterflows6,7,8,9,10. From our daily observation of urban traffic to laboratory experiments, the emergence of counter-propagating 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 materials11,12. However, a description as isotropic grains is usually not sufficient to account for the dynamics of interacting motile bodies13,14,15. From motility-induced phase separation15, to giant density fluctuations in flocks13,16,17, to pedestrian scattering18,19, 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 self-propelled bodies with repulsive interactions, we numerically evidence two non-equilibrium 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 particles20. Finally, we construct a hydrodynamic description to elucidate these long-range structural correlations, and conclude that they are universal to both active and driven ensembles of oppositely moving bodies.
A minimal model of active binary mixtures
We consider an ensemble of N self-propelled particles characterized by their instantaneous positions ri(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 Θi=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 Tij. 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 self-propelled colloids or polymeric biofilaments13,21,22,23,24,25,26,27. 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 centre-to-centre vector rij=(ri−rj): 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(rij)=B(1−rij/(ai+aj)), where B is a finite constant if rij<(ai+aj) 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 ai 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 set-up is used to single out the importance of repulsion torques typical of active bodies. Any more realistic description would also include hard-core interactions. However, in the limit of dilute ensembles and long-range repulsive torques, hard-core interactions are not expected to alter any of the results presented below. Second, unlike models of driven colloids or grains interacting by repulsive forces1,5,20, 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.
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 sweep-and-prune algorithm for neighbour summation. We use a rectangular simulation box of aspect ratio Lx=2Ly 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 self-organize into counter-propagating 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 first-order transition in equilibrium liquids, we now argue that the lane and the mingled states are two genuine non-equilibrium 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 non-zero 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 Bc=2.17±0.02, while above Bc 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−Bc|−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 power-law 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 two-body 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 diameter31. Here these transverse displacements are not bounded by the range of the repulsive interactions. For a finite set of impact parameters, collisions between self-propelled 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 two-body 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.
Long-range correlations in mingled liquids
We now evidence long-range structural correlations in this active-liquid 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 gαβ(x, y=0) decay algebraically as with , Fig. 4b. This power-law behaviour is very close to that reported in numerical simulations4 and fluctuating density functional theories of oppositely driven colloids at finite temperature20.
To explain the robustness of these long-range 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 kinetic-theory framework, for example, refs 32, 33 from an active-matter 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 self-propelled particles align their orientation with the centre-to-centre axis regardless of their initial orientation and external drive. Assuming molecular chaos and binary collisions only, the time evolution of the one-point distribution functions ψα(r, θ, t) reads:
The convective term on the l.h.s stems from self-propulsion, 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 so-called collision integral on the r.h.s., we can establish the dynamical equations for the density fluctuations δρα around the average homogeneous state (see Methods section for technical details). Within a linear response approximation, they take the compact form:
where Jα describes the convection and the collision-induced 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 small-density 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 (self-advected) densities of the two populations34. The associated mass currents are constructed from the only two vectors that can be formed in this homogeneous but anisotropic setting: hα and ∇δρα. 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 mean-field picture we classically account for fluctuations by adding a conserved noise source to equation (5) and compute the resulting density-fluctuation spectrum13. 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 cross-correlation 〈δρα(q)δρβ(−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 spectra20. A key observation is that the structure factor given by equation (8) is non-analytic 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 non-analyticity 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 self-propelled particles, Fig. 4b, and driven colloids4,20. Beyond these long-range 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/x1/2.
Different non-equilibrium 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 non-analytic 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 non-zero. 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 particles20. Self-propulsion causes the particles to move, on average, at constant speed along the x-direction while frontal collisions induce their transverse diffusion. As a result the x-position of the particles increase linearly with time, and their transverse position increases as ∼t1/2. We therefore expect the longitudinal and transverse correlations to be related by a homogeneous function of y/x1/2 in steady state as observed in simulations of both active and driven particles. Altogether these observations confirm the universality of the long-range structural correlations found in both classes of non-equilibrium 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 long-range structural correlations in any ensemble of particles driven towards opposite directions, should they be passive colloids or self-propelled agents.
Boltzmann kinetic theory
Let us summarize the main steps of the kinetic theory employed to establish equations (5)–(7). The so-called 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 two-fluid picture, the velocity and nematic texture of the α particles are given by and . The mass conservation relation, ∂tρα+∇·(ραVα)=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 self-propulsion. 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 repulsion-induced pressure, and the third one echoes the collision-induced rotational diffusivity of the particles. An additional closure relation between Qα, vα and ρα is required to yield a self-consistent hydrodynamic description. Deep in the homogeneous phase, we make a wrapped Gaussian approximation for the orientational fluctuations in each population24,36. 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 vα in the mass conservation relation, leading to the mass conservation equation (5).
The data that support the findings of this study are available from the corresponding author upon request.
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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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.
The authors declare no competing financial interests.
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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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