Stationary distributions of propelled particles as a system with quenched disorder

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