Spatial distributions of nonconservatively interacting particles

Certain types of active systems can be treated as an equilibrium system with excess nonconservative forces driving some of the microscopic degrees of freedom. We derive results for how many particles having both conservative and nonconservative forces will behave. Treating nonconservative forces perturbatively, we show how the probability distribution of the microscopic degrees of freedom is modified from the Boltzmann distribution. We then derive approximate forms of this distribution through analyzing the nature of our perturbations. We compare the perturbative expansion for the microscopic probability distribution to an exactly solvable active system. Finally, we consider how the approximate forms for the microscopic distributions we have derived lead to different macroscopic states when coarse grained for two different kinds of systems, a collection of motile particles, and a system where nonconservative forces are applied in space. In the former, we are able to show that nonconservative forces lead to an effective attractive interaction between motile particles, and in the latter we note that by introducing nonconservative interactions between particles we modify densities through extra terms which couple to surfaces. In this way, we are able to recast certain active problems as the statistical mechanics of nonconservative forces.