Theory for the dynamics of dense systems of athermal self-propelled particles

We present a derivation of a recently proposed theory for the time dependence of density fluctuations in stationary states of strongly interacting, athermal, self-propelled particles. The derivation consists of two steps. First, we start from the equation of motion for the joint distribution of particles' positions and self-propulsions and we integrate out the self-propulsions. In this way we derive an approximate, many-particle evolution equation for the probability distribution of the particles' positions. Second, we use this evolution equation to describe the time dependence of steady-state density correlations. We derive a memory function representation of the density correlation function and then we use a factorization approximation to obtain an approximate expression for the memory function. In the final equation of motion for the density correlation function the nonequilibrium character of the active system manifests itself through the presence of a new steady-state correlation function that quantifies spatial correlations of the velocities of the particles. This correlation function enters into the frequency term, and thus it describes the dependence of the short-time dynamics on the properties of the self-propulsions. More importantly, the correlation function of particles' velocities enters into the vertex of the memory function and through the vertex it modifies the long-time glassy dynamics.