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Universal Long Ranged Correlations in Driven Binary Mixtures

When two populations of"particles"move in opposite directions, like oppositely charged colloids under an electric field or intersecting flows of pedestrians, they can move collectively, forming lanes along their direction of motion. The nature of this"laning transition"is still being debated and, in particular, the pair correlation functions, which are the key observables to quantify this phenomenon, have not been characterized yet. Here, we determine the correlations using an analytical approach based on a linearization of the stochastic equations for the density fields, which is valid for dense systems of soft particles. We find that the correlations decay algebraically along the direction of motion, and have a self-similar exponential profile in the transverse direction. Brownian dynamics simulations confirm our theoretical predictions and show that they also hold beyond the validity range of our analytical approach, pointing to a universal behavior.


Introduction
Binary driven mixtures, in which two populations of particles are driven in opposite directions, encompass a variety of systems, ranging from pedestrian traffic [1][2][3] to charged particle systems like colloidal suspensions [4], electrolytes [5,6], plasmas [7], or ionic liquids [8]. For a large enough driving force, such systems undergo an out-of-equilibrium transition into a state in which the particles moving in the same direction spontaneously form lanes, thereby increasing their average velocity [9,10]. The nature of the laning transition remains unclear [10][11][12][13][14][15][16] and still little is known about the correlations-which are key observables to quantify the transition-and in particular about their decay at large distances [14].
Previous studies suggest that these correlations may be long-ranged. Indeed, when only a small fraction of the particles is biased, the correlation between them and the unbiased particles follows the distribution of the density of particles in the wake of a single intruder moving in a quiescent bath, which is relevant for microrheology [17][18][19][20]. This distribution has been computed for different models: Brownian dilute hard spheres [21], dense soft spheres [22] or hard core particles on a lattice [23]. In the last two cases, the density behind the intruder was found to approach its unperturbed value as a power law.
The situation is by far less clear when half of the particles are driven in one direction. In this case, the correlations have only recently been investigated using numerical simulations and a numerical integration of a closure of the N-particles Fokker-Planck equation [14]. It was shown that the correlations may decay as a power law, with an exponent between 1 and 2, and it was argued that it could be related to a diverging correlation length, and hence, to a true phase transition. However, this observation seemingly contradicts the existence of a finite correlation length, as measured in numerical simulations of large systems [13]. In view of this apparent controversy, it is desirable to have some analytical predictions on the asymptotic decay of correlations. Here, we determine the correlations in a model driven binary mixture using an analytical approach based on a linearized Stochastic Density Functional Theory (SDFT). This approach was introduced recently to compute the correlations in an electrolyte [6] and is reliable for dense systems of soft particles. We obtain that the correlations decay as a power law with an exponent (d + 1)/2 at any value of the external force, in both directions along the field, and have a universal exponential shape in the transverse direction. Brownian dynamics simulations of harmonic spheres agree quantitatively well with our predictions, and also demonstrate that the scaling and the shape of the correlations hold beyond the validity range of our theory. This observation suggests a universal character of the behavior of correlations.

Model
We consider a simple model of a binary driven mixture which comprises N colloidal particles, suspended in a solvent which acts as a heat bath at temperature T , and interacting via an isotropic pair potential V (x). A fraction τ = τ 1 of the particles-called "the tracers" in what follows-is also subject to an external constant force F [10,13,14] (Fig. 1). We assume an overdamped Langevin dynamics so that the coordinates x i (t) ∈ R d of the particles obeẏ where the Gaussian white noise η i (t) has a correlation function η i (t)η j (t ) T = 2T δ ij δ(t −t )1, the mobility is set to κ = 1, T 1 is set of tracers, and T 2 is the set of "non-tracers", present at fraction 1 − τ = τ 2 . This model can also describe situations where the two species are submitted to different external forces, since by symmetry only the difference of the forces applied to the two species affects the correlations [10]. Following Refs. [10,13,14], we do not consider hydrodynamic interactions. This approximation should be valid if the hydrodynamic radius of the particles is smaller than the range of the interaction V (x) [24], or if the hydrodynamic interactions are screened [4,25] (e.g., in electrophoresis).
In order to quantify the cooperative effects, we introduce the effective mobility κ eff of the tracers as ẋ i = κ eff F for i ∈ T 1 . In the absence of interactions, the mobility is κ eff = 1. Using symmetry arguments, one can show that κ eff (τ ) = 1 − (1 − τ )K(τ ), where K(τ ) = K(1 − τ ) (App. A). The factor 1 − τ comes from the density of non-tracers which hinders the motion of the tracers, and K(τ ) describes the cooperativity of the tracers.
The microscopic density of the two species and the pair correlation functions are defined by where α = 1 for tracers, 2 for non-tracers andρ α = τ αρ = τ α N/V, V being the volume of the system. Averaging Eq. (1), we find that K can be expressed via the pair correlation functions as (App. B) the index denotes the direction parallel to F .

Large scale behavior
The Fourier transforms of the correlation functions Eq. (41) are not continuous at 0. The long distance behavior in real space can be obtained by keeping only their singular part (App. D), leading to where Eqs. (11)(12)(13)(14)(15) are our main results. First, they not only yield the algebraic decay with power (d + 1)/2 of the depleted wake of a single tracer [22], but also generalize this result by defining the shape of the wake. At a vanishing tracers density (τ → 0), β → 1 and H + 21 → 0, the decay of h 21 in front of the tracers is exponential (App. E). Second, they show that for a finite density of tracers the algebraic decay holds also for the tracer-non tracer correlation function h 21 in front of the tracers (x → ∞), and for the tracertracer correlation function h 11 . The tracer-tracer correlation is positive along the axis x ⊥ = 0, indicating a propensity for forming lanes.
In Fig. 2a we show the cooperativity as a function ofρ withρ/T and f kept constant. We observe that the simulation results converge to the analytical prediction asρ → ∞, which is the condition of validity of our computation. The cooperativity displays a minimum at τ = 0.5, which is correctly predicted by the theory (see Fig. 2b).
The correlation functions forρ = 2, T = 0.2 and f = 20 are shown in Fig. 3, and are compared to the numerical integration of Eq. (41). Cuts along the transverse direction, rescaled using Eq. (11), are compared to the numerical integration and to the shape (12) in Fig. 4. A very good collapse is found, which confirms the scaling laws, and a quantitative agreement is found for the shape.

Universality
We now show that the scaling (11) of the correlations and their shape (12), hold far beyond the validity regime of our computation, namely a dense and soft suspension. We simulated our system of harmonic x ⊥ |x | −1/2  Fig. 6: a very good collapse is found, and the shape is well described by Eq. (12) with fitted height and width.
These results suggest that the long-range behavior of the correlations, Eqs. (11,12), is universal. It has a simple interpretation as a diffusive process. Replacing x (or −x , depending on the direction of interest) by the time and x ⊥ by the position, Eq. (11) is the solution of a diffusion equation where the source is the second derivative of a Dirac distribution (notably, it is symmetric and its integral is zero). This behavior emanates from simple properties of our system: (i) The long-distance dynamics without external force is diffusive (i.e., the relaxation rate ofφ(k) scales as k 2 for small k); (ii) The tracers have a finite average velocity; (iii) The number of particles is conserved. We conclude that the large scale behavior of the correlations is universal and should be observed in the systems sharing these properties.

Discussion
We have shown that the correlation functions in a simple model of a driven binary mixture decay algebraically in the direction of the external force, and have a self-similar profile in the direction perpendicular to the external force. This shape is universal and should hold in any system which is diffusive at large scales and where the external field induces a finite average velocity of the particles.
Our results shed light on the previous observations. First, we confirm the conjecture of Ref. [14]: the correlation functions are long-ranged with an exponent 3/2 in d = 2, which is compatible with the value reported in Ref. [14]. However, our conclusion is different: (i) this algebraic decay is not associated with the divergence in the structure factors, (ii) it does not indicate a phase transition, instead it is observed over a whole region of the parameter space. Second, in agreement with Ref. [13], we find no trace of a phase transition. However, we show that the correlation length defined in Ref. [13] does not correspond to an exponential decay of the correlations, as is usually the case; resolving this apparent contradiction certainly deserves future work.
In the model considered here, the system is fluid, the dynamics is overdamped and the interactions are short-ranged. However, driven binary mixtures may be very different: they may have an underdamped dynamics [7,30,31] (e.g. in plasmas), or even an active dynamics [3], long-ranged interactions (electrostatic [6] or hydrodynamic), a solid-like behavior in the absence of an external force (e.g., if the system is glassy) [32], etc. It seems natural to inquire next how these differences would affect the large distance behavior of the

Acknowledgments
The authors thank D. S. Dean for stimulating discussions.

A Symmetry arguments for the effective mobility
Here we link the effective mobility at a tracer fraction 1 − τ to the one at a fraction τ . Let F /1 (resp. F /2 ) the force applied to one tracer (resp. one non-tracer) by all the other particles, the effective mobility is Let us now add a force −F on all the particles. This leads to consider the particles 2 as the tracers. But the addition of an external force only translates the system at a constant velocity −F , hence the correlation functions are not modified, and neither do the internal forces (F /1 and F /2 ). The effective mobility as tracer fraction 1 − τ is given by Introducing ∆κ = 1 − κ eff , we get ∆κ(τ ) The third law of Newton for the inter-particle interactions gives: leading to Introducing K(τ ) = ∆κ(τ )/(1 − τ ) allows to rewrite it as

B Effective Mobility From The Correlation Functions
In this section, we prove Eq. (4). Averaging Eq. (1) and summing over the tracers, we get The left-hand side is τ Nκ eff F , hence a projection onto the direction parallel to the force leads to Since the potential is isotropic, ∇V (0) = 0; this allows us to include all the particles in the second sum in the previous equation. The sums can then be rewritten using the density fields ρ α (x): The average can be expressed with the pair correlation functions defined in Eq.
The last equation is obtained from the fact that 1 and h 11 (x) are even functions of x, so that their product with ∂ V (x), which is an odd function of x, has a vanishing integral. Introducing this result in Eq. (24), we get Finally, usingρ 2 = (1 − τ )ρ and the definition of K, leads to Eq. (4):

C Computation of the correlation functions
Our goal is to solveÃC We denoteÃ where and c 11 , c 12 , c 21 and c 22 are the unknowns.
Our system of 4 equations with four unknowns (30) writes: We can solve for c 22 : And the rest follows, Finally, from the values of α, β, γ and δ (33): The behavior of a function at large distances is related to the regularity of its Fourier transform. The fact that the Fourier transform of the correlation function is not continuous at k = 0 (the limits k → 0 and k ⊥ → 0 do not commute) points to an algebraic decay in real space. In order to determine the long range properties of the correlation function, we can extract the singular partC s (k) of its Fourier transformC(k) (Eq. (41)) and compute its inverse Fourier transform. The singular part is defined by the fact thatC(k) −C s (k) is more regular thanC(k) at k = 0, hence the inverse Fourier transform of this difference decays faster than the correlation function in real space. With this definition, the singular partC s (k) is defined up to a regular function. First, the potentialṽ (k) is regular at 0, we can thus replaceṽ (k) byṽ (0) = v 0 in the singular part. This means that at large distances, the interaction potential acts effectively as a Dirac distribution. The terms responsible for the discontinuity are those of the form k /k 2 . Multiplying the numerator and denominator of the correlation function by k 4 , we see that the singular part can be obtained by replacing k 2 = k 2 + k 2 ⊥ by k 2 ⊥ . We thus obtain for the singular part: As we see in the next subsection, the inverse Fourier transform of this function can be computed analytically.

D.2 Inverse Fourier transform of the singular part
We compute the inverse Fourier transform of the singular part of the Fourier transform of the correlation function,C s (k). In the remainder of this section, we drop the subscript "s" inC s for brevity. We are interested inC 21 andC 11 that we writẽ where First, we compute the inverse Fourier transform along the direction of the force, We compute this integral using the residue theorem, leading to The Dirac term in the tracer-tracer correlation function is the contribution of a tracer with itself. It has no effect on the long-distance behavior. The inverse Fourier transform in the transverse direction is straightforward to compute: we recognize the second derivative of a Gaussian. We get Eqs. (11)(12)(13)(14)(15).

E Tracer-non tracer correlation in front of the tracer at vanishing tracer fraction
We focus on the behavior of the tracer-non tracer correlation C 21 (x) for x > 0, in the limit of vanishing tracer fraction, τ → 0. Since we are interested in distances large compared to the range of the potential, we may approximate it by a Dirac: Thus, we want to inverse Fourier transformC where A and α are given by Eqs. (44) and (46). Using k 2 = k 2 + k 2 ⊥ , we can compute the inverse Fourier transform along k with the residue theorem, leading to: where we have introduced q = k 2 ⊥ + f 2 4α 2 1/2 . Since we are interested in the behavior at large x , we can use the saddle-point method to compute the inverse Fourier transform in the transverse direction, leading to The first term in the exponential shows that the decay is exponential. We remark that A is negative, hence the correlation is positive: there is a finite size "traffic jam" in front of the tracers. We also note that the factor [τ (1 − τ )] 1/2 in A, which goes to zero when τ → 0, is removed if we consider the pair correlation function h 21 (x). The convergence of the average force on the tracers starting from the initial random configuration is shown in Fig. 7 for 10 different simulations); it shows that the stationnary state is reached after 2 × 10 4 iterations. For each set of parameters, the observables are then averaged over a set of 50 simulations.

F.2 Code
The code used in our simulations is released as a free software under CeCILL 2.1 license [29].

F.3 Finite size effects
In order to check that finite size effects are absent in our numerical simulations, we have run the simulations for the parameters used in Figs. 3 and 4, for different system sizes: N ∈ {1, 2, 5, 10, 20, 40, 80} × 10 3 .
In Fig. 8, we show the decay of the correlation function C 12 in the direction of F , i.e., x ⊥ = 0. Finite size effects are indeed present, but they show up at larger and larger distances as the system size increases. In Fig. 9, the transverse profile of the correlation function C 12 is shown for x = 10; it depends on the size of the system but it converges as the size increases.
In the dilute and hard regime (Figs. 5 and 6), we can use the results of Glanz and Löwen [13], who used similar parameters (albeit with a different interaction potential V (x)). They found that for N 8 · 10 4 , finite size effects may be expected (i.e., the correlation length becomes comparable to the size of the system) for Péclet numbers f 120, which is well above the Péclet number that we used, f = 20.