Active-Passive Brownian Particle in Two Dimensions

TL;DR

This paper introduces a two-dimensional active-passive Brownian particle model with stochastic speed fluctuations, bridging active and passive Brownian motion, and providing analytical expressions for displacement moments and effective diffusivity.

Contribution

It relaxes the constant speed assumption in active Brownian motion, allowing stochastic speed variations and establishing a connection to run-and-tumble particle models.

Findings

Derived analytical expressions for displacement moments.

Calculated effective diffusion coefficient.

Mapped run-and-tumble particles onto the model.

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.