Confined run-and-tumble particles with non-Markovian tumbling statistics

Confined active particles constitute simple, yet realistic, examples of systems that converge into a nonequilibrium steady state. We investigate a run-and-tumble particle in one spatial dimension, trapped by an external potential, with a given distribution $g\left(t\right)$ of waiting times between tumbling events whose mean value is equal to $\tau$. Unless $g\left(t\right)$ is an exponential distribution (corresponding to a constant tumbling rate), the process is non-Markovian, which makes the analysis of the model particularly challenging. We use an analytical framework involving effective position-dependent tumbling rates to develop a numerical method that yields the full steady-state distribution (SSD) of the particle's position. The method is very efficient and requires modest computing resources, including in the large-deviation and/or small-$\tau$ regime, where the SSD can be related to the the large-deviation function, $s\left(x\right)$, via the scaling relation $P_{\mathrm{st}}\left(x\right)\sim e^{-s\left(x\right)/\tau }$.

Figure 1

The function, $s_{\tau }\left(x\right)$ [which is related to the SSD $P\left(x\right)$ via Eq. (29)] for to an exponential distribution of running times, $g\left(t\right)=exp(-t/\tau )/\tau$, for $\tau =1$ (black circles) and $\tau =10$ (blue squares). The red line shows the analytical solution (30). The inset is a magnification of the central region $0<x<0.9$.

Figure 2

The dimensionless local switching rate, $\tau {\gamma }_{+}\left(x\right)$, corresponding to an exponential distribution of running times, for $\tau =1$ (black solid line) and $\tau =10$ (blue dashed line).

Figure 3

The function $s_{\tau }\left(x\right)$ [which is related to the SSD $P\left(x\right)$ via Eqs. (9) and (28)] corresponding to a semi-Gaussian distribution of running times [Eq. (31)] for $\tau =1$ (black circles) and $\tau =10$ (blue squares). The inset is a magnification of the central region $0<x<0.9$, with data also corresponding to $\tau =0.1$ which is plotted with red triangles.

Figure 4

The dimensionless local switching rate, $\tau {\gamma }_{+}\left(x\right)$ corresponding to a semi-Gaussian distribution of running times, for $\tau =0.1$ (dashed-dotted red line), $\tau =1$ (black solid line), and $\tau =10$ (blue dashed line).

Figure 5

The SSD, $P\left(x\right)$, corresponding to a half-$t$ distribution of running times [Eq. (36)] for $\tau =0.1$ (dashed-dotted red line), $\tau =1$ (black solid line), and $\tau =10$ (blue dashed line).

Figure 6

The dimensionless local switching rate, $\tau {\gamma }_{+}\left(x\right)$ corresponding to a half-$t$ distribution of running times, for $\tau =0.1$ (dashed-dotted red line), $\tau =1$ (black solid line), and $\tau =10$ (blue dashed line).