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 $u_{\mathrm{eff}}$. 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 $e^{-\beta u_{\mathrm{eff}}}$. In the case of a run-and-tumble model, the formulation reveals an effective interaction between particles.

Figure 1

Distributions of propelled particles in the potential $u_{\mathrm{ext}}=K{r}^2/2$ obtained from dynamic simulations for $d=2$ (dashed black lines). ${\lambda }_k=\sqrt{2/\beta K}$ is the trap size, and the results are for $v_0{\lambda }_k/D=5$. The solid red line corresponds to the expression in (10). The results in (a) are for RTP and those in (b) for ABP type of motion.

Figure 2

Distributions of propelled particles between two parallel walls separated by $2h$ obtained from dynamic simulations for $d=2$ (dashed black lines). The results are for $v_{0}h/D=10$. The solid red line corresponds to the distribution ${\overline{n}}_0\left(x\right)$ in the quenched disorder limit. The results in (a) are for RTP motion and (b) for ABP motion.

Figure 3

Distributions of propelled particles in the external potential $u_{\mathrm{ext}}=K{x}^2/2$ obtained from dynamic simulations for $d=2$ (dashed black lines). ${\lambda }_k=\sqrt{2/\beta K}$ is the trap size and the results are for $v_0{\lambda }_k/D=5$. The solid red line corresponds to the distribution ${\overline{n}}_0\left(x\right)$. The results in (a) are for RTP and (b) for ABP motion.

Figure 4

Distributions ${\overline{n}}_0\left(x\right)$ for an external potential (a) $u_{\mathrm{ext}}=K{x}^2/2$ and (b) for particles between two walls, for different system dimensionality $d$. The distributions are for $v_0{\lambda }_k/D=5$ and $v_{0}h/D=10$.

Figure 5

Distributions $\overline{n}\left(x\right)$ obtained numerically using the SC formulation of the FP equation for the RTP particles between two walls for $d=2$ and $v_{0}h/D=10$. Circles correspond to simulation data points.

Figure 6

Distributions $\overline{n}\left(x\right)$ obtained numerically using the SC formulation for the RTP particles in the potential $u_{\mathrm{ext}}=K{x}^2/2$ for $d=2$ and $v_0{\lambda }_k/D=5$. Circles correspond to simulation data points.

Figure 7

Distributions $\overline{n}\left(x\right)$ obtained numerically from the SC formulation for the ABP particles for $d=2$ (a) between two walls with $v_{0}h/D=10$ and (b) in the harmonic potential with $v_0{\lambda }_k/D=5$. Circles correspond to simulation data points.