Stationary distributions of propelled particles as a system with quenched disorder

This article is the exploration of the viewpoint within which propelled particles in a steady-state are regarded as a system with quenched disorder. The analogy is exact when the rate of the drift orientation vanishes and the linear potential, representing the drift, becomes part of an external potential, resulting in the effective potential $u_{eff}$. The stationary distribution is then calculated as a disorder-averaged quantity by considering all contributing drift orientations. To extend this viewpoint to the case when a drift orientation evolves in time, we reformulate the relevant Fokker-Planck equation as a self-consistent relation. One interesting aspect of this formulation is that it is represented in terms of the Boltzmann factor $e^{-\beta u_{eff}}$. In the case of a run-and-tumble model, the formulation reveals an effective interaction between particles.