Active Brownian particle

Mathematically, an active Brownian particle is described by its center of mass coordinates ${\displaystyle \mathbf {r} }$ and a unit vector ${\displaystyle {\hat {\mathbf {n} }}}$ giving the orientation. In two dimensions, the orientation vector can be parameterized by the 2D polar angle ${\displaystyle \theta }$, so that ${\displaystyle {\hat {\mathbf {n} }}=(\cos \theta ,\sin \theta )}$. The equations of motion in this case are the following stochastic differential equations:

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

where

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

with ${\displaystyle \mathbf {I} }$ the 2×2 identity matrix. The terms ${\displaystyle {\boldsymbol {\eta }}_{\text{trans}}(t)}$ and ${\displaystyle \eta _{\text{rot}}(t)}$ are translational and rotational white noise, which is understood as a heuristic representation of the Wiener process. Finally, ${\displaystyle V(\mathbf {r} )}$ is an external potential, ${\displaystyle m}$ is the mass, ${\displaystyle \xi }$ is the friction, ${\displaystyle v_{0}}$ is the magnitude of the self-propulsion velocity, and ${\displaystyle D}$ and ${\displaystyle D_{t}}$ are the translational and rotational diffusion coefficients.[6]

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

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

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 ${\displaystyle \ell =v_{0}/D_{r}}$, called the persistence length.[1]

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.[7][8]

Sufficiently concentrated suspensions of active Brownian particles phase separate into a dense and dilute regions.[9][10] The particles' motility drives a positive feedback loop, in which particles collide and hinder each other's motion, leading to further collisions and particle accumulation.[1] At a 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.[9] It is a type of athermal phase separation because it occurs even if the particles are spheres with hard-core (purely repulsive) interactions.

A variant of active Brownian motion involves complete directional reversals in addition to rotational diffusion. This movement pattern is seen in bacteria like Myxococcus xanthus, Pseudomonas putida, Pseudoalteromonas haloplanktis, Shewanella putrefaciens, and Pseudomonas citronellolis.[11]

• Run-and-tumble motion
• Janus particle
• Langevin equation
• Active matter
• Nonequilibrium statistical mechanics