Dynamical self-consistent mean-field theory for interacting self-propelled rods

We derive a dynamical self-consistent field theory (dSCFT) for interacting, self-propelled rods by applying an extremal principle to the collective variables in an exact Martin-Siggia-Rose functional integral arising from the microscopic equations for the many-body Brownian dynamics of self-propelled rods. In dSCFT, the problem reduces to that of the motion of one self-propelled rod under the influence of dynamical forces and torques from other rods computed within a self-consistent mean-field approximation. A key quantity in the theory is the time-dependent rod position-orientation distribution function, that satisfies a Smoluchowski equation. Our derivation provides a clean, systematic, and direct route from the microscopic equations of rod motion to this continuum dynamical theory. This connection will enable us to efficiently and quantitatively simulate dSCFTs for biological systems of increasing complexity and realism. We present simulations that aim to understand the fascinating large-scale collective structures and group dynamics seen in studies of the surface motility of Pseudomonas aeruginosa bacteria colonies.