Stationary distributions of propelled particles as a system with quenched disorder

TL;DR

This paper investigates the stationary distributions of propelled particles viewed as a system with quenched disorder, introducing a reformulation of the Fokker-Planck equation and revealing effective interactions in run-and-tumble models.

Contribution

It provides a novel perspective by modeling propelled particles as a quenched disorder system and reformulates the Fokker-Planck equation to include time-evolving drift orientations.

Findings

Stationary distribution as a disorder-averaged quantity.

Reformulation of Fokker-Planck equation as a self-consistent relation.

Effective interactions in run-and-tumble particle models.

Abstract

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.