Power functional theory for active Brownian particles: General formulation and power sum rules

We generalize power functional theory Schmidt and Brader, J. Chem. Phys. 138, 214101 (2013) to Brownian many-body systems with orientational degrees of freedom. The framework allows the study of active particles in general inhomogeneous and time-dependent nonequilibrium. We prove for steady states that the free power equals half the negative dissipated external work per time, and is hence trivially related to the average forward swim speed of the particles. The variational theory expresses the free power as a functional of the microscopic one-body density and current distribution. Both fields are time-, position- and orientation-dependent, and the total current consists of translational and rotational parts. Minimization of the free power functional with respect to the current(s) yields the physical dynamics of the system. We give a simple approximation for the superadiabatic (above adiabatic) contribution which describes excess dissipation in homogeneous bulk fluids due to drag. In steady states, we evaluate the free power using Brownian dynamics simulations for short-ranged soft repulsive spheres. We describe the necessary sampling strategies and show that the theory provides a good account of the simulation data.