Active-Passive Brownian Particle in Two Dimensions

TL;DR

This work studies a two-dimensional active Brownian particle whose self-propulsion speed switches stochastically between an active value $s$ and a passive value $0$ via a random telegraph process with rates $\alpha$ and $\beta$. By solving the Langevin dynamics for translational and rotational diffusion, the authors obtain exact expressions for the first two displacement moments and derive an analytical form for the long-time diffusion coefficient $D_{\mathrm{eff}} = D_t + \frac{s^2}{2D_r} \frac{\alpha}{(\alpha+\beta)^2} \left( \alpha + \beta \frac{D_r}{\alpha+\beta+D_r} \right)$. They further demonstrate that a run-and-tumble particle can be matched to this model in the sense that their large-scale diffusivities coincide, via an implicit relationship between the switching rates and tumble rate. Overall, the active–passive Brownian particle provides a compact, analytically tractable framework that interpolates between passive Brownian motion, active Brownian motion, and run-and-tumble dynamics, offering insights for biological and synthetic systems with intermittent motility. $D_{\mathrm{eff}}$ expressions and moment formulas enable direct comparisons to experimental data and model reductions to established motility paradigms.

Abstract

We describe a two-dimensional model for active particles whose self-propulsion speed is not fixed, but varies in time, and whose motion is subject to both translational and rotational diffusion. In the conventional treatment of active Brownian motion, the self-propulsion speed is taken to be constant - an assumption convenient for analysis but poorly matched to many real systems. Here we relax that assumption, allowing the speed $v(t)$ to fluctuate stochastically between two values: $v=0$ (a passive state) and $v=s$ (an active state). Transitions between these states are taken to follow a random telegraph process. This ``active-passive Brownian particle'' inherits limiting behaviors from both the purely active and purely passive Brownian cases. Analytical expressions for the first two displacement moments, and for the resulting effective diffusion coefficient, make this dual character explicit. Moreover, by an appropriate identification of parameters, a run-and-tumble particle - such as a motile bacterium - can be mapped onto this model in such a way that their large-scale diffusivities coincide.

Figures (4)

• Figure 1: A schematic active–passive Brownian particle: the small blue disk is at position ${\mathbf{r}}$ and moves in the unit direction ${\mathbf{u}}$ set by the orientation angle $\varphi$ relative to the $x_1$-axis.
• Figure 2: Mean displacement $\langle {\mathbf{r}}(t)-{\mathbf{r}}_0 \rangle$ as a function of time for a particle of radius $R=1 \,\mu$m in water at room temperature, with $D_t=0.24\, \mu\mathrm{m}^2/\mathrm{s}$, $\alpha=\beta=D_r=0.18\,\mathrm{s}^{-1}$, and $s=3 \,\mu\mathrm{m}/\mathrm{s}$. The two curves correspond to starting from rest ($v_0=0$) and starting active ($v_0=s$). The figure illustrates the short-time quadratic versus linear growth and the eventual saturation predicted by Eq. (\ref{['eq: average of r(t)']}).
• Figure 3: Mean-square displacement for the same parameters as in Fig. \ref{['fig: MeanPosition']}. At short times, the MSD grows linearly, reflecting the initial diffusive regime. At intermediate times, nonlinear behavior emerges as the interplay between switching dynamics $(\alpha+\beta)^{-1}$ and rotational diffusion $D_r^{-1}$ becomes significant. At long times, the MSD returns to linear growth, now characterized by the effective diffusion coefficient $D_{eff}$ given by Eq. (\ref{['eq: effective diffusion coefficient']}).
• Figure 4: Mean-square displacement when $D_t=0$, all other parameters as in Fig. \ref{['fig: MeanPosition']}. With translational diffusion suppressed, the short-time linear regime disappears, and intermediate-time ballistic or superdiffusive behavior becomes prominent, as anticipated from the analysis of Eq. (\ref{['eq: MSD']}).