Active Brownian particle

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An active Brownian particle (ABP) is a model of self-propelled motion in a dissipative environment.[1][2] It is a nonequilibrium generalization of a Brownian particle. The self-propulsion results from a force that acts on the particle's center of mass and points in the direction of an intrinsic body axis (the particle orientation).[2] It is common to treat particles as spheres, though other shapes (such as rods) have also been studied.[3][4] Both the center of mass and the direction of the propulsive force are subjected to white noise, which contributes a diffusive component to the overall dynamics. In its simplest version, the dynamics is 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 and its straightforward extensions, though some authors have used it for more general self-propelled particle models.[4][5]

A special variant of active Brownian motion is seen in bacteria like Myxococcus xanthus, Pseudomonas putida, Pseudoalteromonas haloplanktis, Shewanella putrefaciens, Pseudomonas citronellolis, where they undergo complete directional reversals in addition to the usual active Brownian motion like dynamics. They are popularly called direction reversing active Brownian particles (DRABP)[6].

Equations of motion

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

[math]\displaystyle{ \begin{align} \dot{\mathbf{r}} &= v_0 \hat{\mathbf{n}} - (m \xi)^{-1} \nabla V(\mathbf{r}) + \sqrt{2 D} \, \boldsymbol{\eta}_{\text{trans}}(t) \\ \dot{\theta} &= \sqrt{2 D_r} \, \eta_{\text{rot}}(t). \end{align} }[/math]


[math]\displaystyle{ \begin{align} \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}^{\intercal}_{\text{trans}}(t') \rangle = \mathbf{I} \delta(t-t') \end{align} }[/math]

with [math]\displaystyle{ \mathbf{I} }[/math] the 2×2 identity matrix. The terms [math]\displaystyle{ \boldsymbol{\eta}_{\text{trans}}(t) }[/math] and [math]\displaystyle{ \eta_{\text{rot}}(t) }[/math] are translational and rotational white noise, which is understood as a heuristic representation of the Wiener process. Finally, [math]\displaystyle{ V(\mathbf{r}) }[/math] is an external potential, [math]\displaystyle{ m }[/math] is the mass, [math]\displaystyle{ \xi }[/math] is the friction, [math]\displaystyle{ v_0 }[/math] is the magnitude of the self-propulsion velocity, and [math]\displaystyle{ D }[/math] and [math]\displaystyle{ D_t }[/math] are the translational and rotational diffusion coefficients.[7]

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

[math]\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 }[/math]


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 [math]\displaystyle{ \ell = v_0/D_r }[/math], 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.[8][9]

Sufficiently concentrated suspensions of active Brownian particles phase separate into a dense and dilute regions.[10][11] 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.[10] It is a type of athermal phase separation because it occurs even if the particles are spheres with hard-core (purely repulsive) interactions.

See also



  • Baek, Yongjoo; Solon, Alexandre P.; Xu, Xinpeng; Nikola, Nikolai; Kafri, Yariv (2018-01-31). "Generic Long-Range Interactions Between Passive Bodies in an Active Fluid". Physical Review Letters (American Physical Society (APS)) 120 (5). doi:10.1103/physrevlett.120.058002. ISSN 0031-9007. 
  • Bechinger, Clemens; Di Leonardo, Roberto; Löwen, Hartmut; Reichhardt, Charles; Volpe, Giorgio; Volpe, Giovanni (2016-11-23). "Active Particles in Complex and Crowded Environments". Reviews of Modern Physics (American Physical Society (APS)) 88 (4). doi:10.1103/revmodphys.88.045006. ISSN 0034-6861. 
  • Cates, Michael E.; Tailleur, Julien (2015-03-01). "Motility-Induced Phase Separation". Annual Review of Condensed Matter Physics (Annual Reviews) 6 (1): 219–244. doi:10.1146/annurev-conmatphys-031214-014710. ISSN 1947-5454. 
  • Fodor, Étienne; Cristina Marchetti, M. (2018). "The statistical physics of active matter: From self-catalytic colloids to living cells". Physica A: Statistical Mechanics and its Applications (Elsevier BV) 504: 106–120. doi:10.1016/j.physa.2017.12.137. ISSN 0378-4371. 
  • Marchetti, M. Cristina; Fily, Yaouen; Henkes, Silke; Patch, Adam; Yllanes, David (2016). "Minimal model of active colloids highlights the role of mechanical interactions in controlling the emergent behavior of active matter". Current Opinion in Colloid & Interface Science (Elsevier BV) 21: 34–43. doi:10.1016/j.cocis.2016.01.003. ISSN 1359-0294. 
  • Ni, Ran; Cohen Stuart, Martien A.; Bolhuis, Peter G. (2015-01-07). "Tunable Long Range Forces Mediated by Self-Propelled Colloidal Hard Spheres". Physical Review Letters (American Physical Society (APS)) 114 (1). doi:10.1103/physrevlett.114.018302. ISSN 0031-9007. 
  • Peruani, Fernando (2016). "Active Brownian rods". The European Physical Journal Special Topics (Springer Science and Business Media LLC) 225 (11-12): 2301–2317. doi:10.1140/epjst/e2016-60062-0. ISSN 1951-6355. 
  • Romanczuk, P.; Bär, M.; Ebeling, W.; Lindner, B.; Schimansky-Geier, L. (2012). "Active Brownian particles". The European Physical Journal Special Topics (Springer Science and Business Media LLC) 202 (1): 1–162. doi:10.1140/epjst/e2012-01529-y. ISSN 1951-6355. 
  • Shaebani, M. Reza; Wysocki, Adam; Winkler, Roland G.; Gompper, Gerhard; Rieger, Heiko (2020-03-10). "Computational models for active matter". Nature Reviews Physics (Springer Science and Business Media LLC) 2 (4): 181–199. doi:10.1038/s42254-020-0152-1. ISSN 2522-5820. 
  • Zöttl, Andreas; Stark, Holger (2016-05-11). "Emergent behavior in active colloids". Journal of Physics: Condensed Matter (IOP Publishing) 28 (25): 253001. doi:10.1088/0953-8984/28/25/253001. ISSN 0953-8984.