Enhanced Diffusion and Ordering of Self-Propelled Rods

Starting from a minimal physical model of self-propelled hard rods on a substrate in two dimensions, we derive a modified Smoluchowski equation for the system. Self-propulsion enhances longitudinal diffusion and modifies the mean-field excluded volume interaction. From the Smoluchowski equation we obtain hydrodynamic equations for rod concentration, polarization and nematic order parameter. New results at large scales are a lowering of the density of the isotropic-nematic transition and a strong enhancement of boundary effects in confined self-propelled systems.

Figure 1

A collision of two self-propelled hard rods (the width of the rod is exaggerated for clarity). $\widehat{\mathbf{k}}$ is a unit vector from rod 2 to rod 1 normal to the point of contact. Points on the side of the rods are identified by vectors ${\xi }_i$.

Figure 2

The polarization in a channel of width $L$ for $\delta /L=0.2$ (solid) and $\delta /L=0.6$ (dashed).