Development of Dynamical Self-Consistent Field-Theory for Active Rods and its Application to Finger-Like Pattern Formation in Bacteria Colonies at a Glass-Agar Interface

We present a theoretical framework for studying dense systems of interacting, active, rod-like particles in two dimensions by modifying the polymer dynamical self-consistent field theory (dSCFT). dSCFT provides a systematic method to derive the Smoluchowski equations seen in the literature for active rod-like particles by taking a saddle-point approximation to the functional integral representation of the microscopic Langevin equations with respect to the collective fields. The microscopic action is easily related to the physical characteristics of the system being studied keeping the theory grounded in reality. The derivation of the Smoluchowski equation, and the form of the collective field variables is the main result of this paper. As a demonstration of the theory we numerically reproduce the disorder-nematic phase transition of hard rods predicted by Onsager, and investigate the effect of adding a self-propulsive force to the system.The interactions between rods are an important aspect of the theory and are treated using a Weeks-Chandler-Anderson interaction potential, where the overlapping cases are appropriately remapped. Under this implementation of the interaction force we see an onset of nematic ordering by increasing the density of rods.