The dynamic structure factor of active Brownian particles

The classical problem of diffusion from a point source for a passive system is known to behave in a purely diffusive fashion. Experiments suggest that for active systems a wavelike character is observed. We study the dynamic structure factor of active Brownian particles (ABPs) with no initial net polar order for the case of ABPs released from a point source with no net nematic order in the system. The transient behavior is captured by the Smoluchowski equation for the evolution of the particle probability density function in orientation and position space. Using a moments expansion method we obtain a wave equation for the number density of ABPs. The number density is characterized by wavelike behavior at short times and becomes diffusive at times long compared to the reorientation time. The diffusive behavior is characterized by an effective diffusivity, which is the sum of the translational diffusivity and swim diffusivity D_swim = U₀² τ_R / 2, where U₀ is the speed of the particle and τ_R is the reorientation time. Our results are corroborated by Brownian dynamic simulations.