Active Brownian particle

An active Brownian particle (ABP) is a model of

nonequilibrium generalization of a Brownian particle

The self-propulsion results from a force that acts on the particle's

overdamped and the propulsive force has constant magnitude, so that the magnitude of the velocity is likewise constant (speed-up to terminal velocity

is instantaneous).

The term active Brownian particle usually refers to this simple model [1] and its straightforward extensions, though some authors have used it for more general self-propelled particle models.^[5]^[6]

Equations of motion

Mathematically, an active Brownian particle is described by its center of mass coordinates $\mathbf {r}$ and a unit vector ${\hat {\mathbf {n} }}$ giving the orientation. In two dimensions, the orientation vector can be parameterized by the

2D polar angle

\theta

, so that

{\hat {\mathbf {n} }}=(\cos \theta ,\sin \theta )

. The equations of motion in this case are the following

stochastic differential equations

:

{\begin{aligned}{\dot {\mathbf {r} }}&=v_{0}{\hat {\mathbf {n} }}-(m\xi )^{-1}\nabla V(\mathbf {r} )+{\sqrt {2D_{t}}}\,{\boldsymbol {\eta }}_{\text{trans}}(t)\\{\dot {\theta }}&={\sqrt {2D_{r}}}\,\eta _{\text{rot}}(t).\end{aligned}}

where

{\begin{aligned}\langle \eta _{\text{rot}}(t)\rangle &=0;\qquad \langle \eta _{\text{rot}}(t)\eta _{\text{rot}}(t')\rangle =\delta (t-t')\\\langle {\boldsymbol {\eta }}_{\text{trans}}(t)\rangle &={\boldsymbol {0}};\qquad \langle {\boldsymbol {\eta }}_{\text{trans}}(t){\boldsymbol {\eta }}_{\text{trans}}^{\intercal }(t')\rangle =\mathbf {I} \delta (t-t')\end{aligned}}

with $\mathbf {I}$ the 2×2 identity matrix. The terms ${\boldsymbol {\eta }}_{\text{trans}}(t)$ and $\eta _{\text{rot}}(t)$ are translational and rotational white noise, which is understood as a heuristic representation of the Wiener process. Finally, $V(\mathbf {r} )$ is an external potential, $m$ is the mass, $\xi$ is the friction, $v_{0}$ is the magnitude of the self-propulsion velocity, and $D_{t}$ and $D_{r}$ are the translational and rotational

diffusion coefficients.^[7]

The dynamics can also be described in terms of a probability density function $f(\mathbf {r} ,\theta ,t)$ , which gives the probability, at time $t$ , of finding a particle at position $\mathbf {r}$ and with orientation $\theta$ . By averaging over the stochastic trajectories from the equations of motion, $f(\mathbf {r} ,\theta ,t)$ can be shown to obey the following partial differential equation:

{\frac {\partial f}{\partial t}}+v_{0}{\hat {n}}\cdot \nabla f=(m\xi )^{-1}\nabla \cdot (\nabla V(\mathbf {r} )\,f)+D_{r}{\frac {\partial ^{2}f}{\partial \theta ^{2}}}+D_{t}\nabla ^{2}f

Behavior

For an isolated particle far from boundaries, the combination of diffusion and self-propulsion produces a stochastic (fluctuating) trajectory that appears ballistic over short length scales and diffusive over large length scales. The transition from ballistic to diffusive motion is defined by a characteristic length $\ell =v_{0}/D_{r}$ , called the persistence length.^[2]

In the presence of boundaries or other particles, more complex behavior is possible. Even in the absence of attractive forces, particles tend to accumulate at boundaries. Obstacles placed within a bath of active Brownian particles can induce long-range density variations and nonzero currents in steady state.^[8]^[9]

Sufficiently concentrated suspensions of active Brownian particles phase separate into a dense and dilute regions.

coarse-grained level, a particle's effective self-propulsion velocity decreases with increased density, which promotes clustering. In the more general context of self-propelled particle models, this behavior is known as motility-induced phase separation.^[10] It is a type of athermal phase separation

because it occurs even if the particles are spheres with hard-core (purely repulsive) interactions.