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.

Figure 1

Snapshot of numerical simulations of Brownian harmonic spheres with $\overline{\rho }=0.5$, $\tau =0.5$, $T={10}^{-4}$, and $F=0.005$. The tracers (red) are subject to an external force $\mathit{F}$ while the nontracers (blue) are not. Particles of the same species tend to form lanes in the direction of the field.

Figure 2

Cooperativity $K$ in $d=2$ as a function (a) of $\overline{\rho }$ with $\overline{\rho }/T=10$, $F/T=20$, and $\tau =0.5$, (b) of the fraction of tracers $\tau$ for $\overline{\rho }=2$, $\overline{\rho }/T=10$, and $F/T=20$. Solid lines and points denote analytical and simulation results, respectively.

Figure 3

Pair correlation functions from the simulations (lower half) and the theory (upper half) for $\overline{\rho }=2$, $T=0.2$, $F=4$, and $\tau =0.5$, in dimension $d=2$. The black circle sets the size of a particle. (a) Tracer–nontracer correlation. (b) tracer-tracer correlation.

Figure 4

Rescaled tranverse profiles of the pair correlation functions for $\overline{\rho }=2$, $T=0.2$, $F=4$, $\tau =0.5$, in dimension $d=2$. Left panel: profiles from the simulations at different longitudinal positions and universal shape Eq. (12) (thick black line). Right panel: profiles at $x_{\parallel }=-10$ from the simulations (solid line) and the analytical prediction obtained by numerical inversion of Eq. (9) (dashed line). (a) Tracer–nontracer correlation. (b) Tracer-tracer correlation.

Figure 5

Pair correlation functions from the simulations for $\overline{\rho }=0.2$, $T=0.001$, $F=0.02$, $\tau =0.5$, in dimension $d=2$. The black circle sets the size of a particle. (a) Tracer–nontracer correlation. (b) Tracer-tracer correlation.

Figure 6

Rescaled tranverse profiles of the pair correlation functions for $\overline{\rho }=0.2$, $T=0.001$, $F=0.02$, $\tau =0.5$, in dimension $d=2$. The transverse profiles from the simulations at different longitudinal positions are shown together with the universal shape Eq. (12) with fitted height and width (thick black line). (a) Tracer–nontracers correlation. (b) Tracer-tracer correlation.