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Active Brownian particle
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<h2 id="equations-of-motion">Equations of motion</h2> <p>Mathematically, an active Brownian particle is described by its center of mass coordinates r {\displaystyle \mathbf {r} } and a <a href="/facts/Unit_vector/GyaXneDn">unit vector</a> n ^ {\displaystyle {\hat {\mathbf {n} }}} giving the orientation. In two dimensions, the orientation vector can be parameterized by the <a href="/facts/2D_polar_angle/HBko4YoN">2D polar angle</a> θ {\displaystyle \theta } , so that n ^ = ( cos θ , sin θ ) {\displaystyle {\hat {\mathbf {n} }}=(\cos \theta ,\sin \theta )} . The equations of motion in this case are the following <a href="/facts/Stochastic_differential_equations/ol9a9WNq">stochastic differential equations</a>: </p> r ˙ = v 0 n ^ − ( m ξ ) − 1 ∇ V ( r ) + 2 D t η trans ( t ) θ ˙ = 2 D r η rot ( t ) . {\displaystyle {\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}}} <p>where </p> ⟨ η rot ( t ) ⟩ = 0 ; ⟨ η rot ( t ) η rot ( t ′ ) ⟩ = δ ( t − t ′ ) ⟨ η trans ( t ) ⟩ = 0 ; ⟨ η trans ( t ) η trans ⊺ ( t ′ ) ⟩ = I δ ( t − t ′ ) {\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}}} <p>with I {\displaystyle \mathbf {I} } the 2×2 <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>. The terms η trans ( t ) {\displaystyle {\boldsymbol {\eta }}_{\text{trans}}(t)} and η rot ( t ) {\displaystyle \eta _{\text{rot}}(t)} are translational and rotational <a href="/facts/White_noise/wJ0SR21n">white noise</a>, which is understood as a heuristic representation of the <a href="/facts/Wiener_process/KPh7vYd7">Wiener process</a>. Finally, V ( r ) {\displaystyle V(\mathbf {r} )} is an external <a href="/facts/Potential_energy/I03KLbmn">potential</a>, m {\displaystyle m} is the mass, ξ {\displaystyle \xi } is the friction, v 0 {\displaystyle v_{0}} is the magnitude of the self-propulsion velocity, and D t {\displaystyle D_{t}} and D r {\displaystyle D_{r}} are the translational and rotational <a href="/facts/Diffusion_coefficient/GSngNem0">diffusion coefficients</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p><p>The dynamics can also be described in terms of a <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> f ( r , θ , t ) {\displaystyle f(\mathbf {r} ,\theta ,t)} , which gives the probability, at time t {\displaystyle t} , of finding a particle at position r {\displaystyle \mathbf {r} } and with orientation θ {\displaystyle \theta } . By averaging over the stochastic trajectories from the equations of motion, f ( r , θ , t ) {\displaystyle f(\mathbf {r} ,\theta ,t)} can be shown to obey the following <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equation</a>: </p> ∂ f ∂ t + v 0 n ^ ⋅ ∇ f = ( m ξ ) − 1 ∇ ⋅ ( ∇ V ( r ) f ) + D r ∂ 2 f ∂ θ 2 + D t ∇ 2 f {\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} <h2 id="behavior">Behavior</h2> <p class="note">See also: <a href="/facts/Clustering_of_self-propelled_particles/752Y7yd9">Clustering of self-propelled particles</a></p> <p>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 ℓ = v 0 / D r {\displaystyle \ell =v_{0}/D_{r}} , called the persistence length.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p><p>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.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p><p>Sufficiently concentrated suspensions of active Brownian particles phase separate into a dense and dilute regions.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> The particles' motility drives a <a href="/facts/Positive_feedback_loop/eFGatCpF">positive feedback loop</a>, in which particles collide and hinder each other's motion, leading to further collisions and particle accumulation.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> At a <a href="/facts/Coarse-grained/kBqMRzaj">coarse-grained</a> level, a particle's <i>effective</i> 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 <i>motility-induced phase separation</i>.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> It is a type of athermal <a href="/facts/Phase_separation/9AQPClRu">phase separation</a> because it occurs even if the particles are spheres with hard-core (purely repulsive) interactions. </p> <h2 id="variations">Variations</h2> <p>A variant of active Brownian motion involves complete directional reversals in addition to rotational diffusion. This movement pattern is seen in bacteria like <i><a href="/facts/Myxococcus_xanthus/Q0LQcmKe">Myxococcus xanthus</a></i>, <i><a href="/facts/Pseudomonas_putida/AaRGu0KL">Pseudomonas putida</a>,</i> <i><a href="/facts/Pseudoalteromonas_haloplanktis/BsuUrDmq">Pseudoalteromonas haloplanktis</a></i>, <i><a href="/facts/Shewanella_putrefaciens/HqpADgcK">Shewanella putrefaciens</a></i>, and <i><a href="/facts/Pseudomonas_citronellolis/zhTxwhh2">Pseudomonas citronellolis</a>.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a></i> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Run-and-tumble_motion/Tkgkx7uc">Run-and-tumble motion</a> – Type of bacterial motion</li> <li><a href="/facts/Janus_particle/FsHakVKt">Janus particle</a> – Type of nanoparticle or microparticlePages displaying short descriptions of redirect targets</li> <li><a href="/facts/Langevin_equation/XPDoCcVI">Langevin equation</a> – Stochastic differential equation</li> <li><a href="/facts/Active_matter/jH9Ph4Yq">Active matter</a> – Matter behavior at system scale</li> <li><a href="/facts/Nonequilibrium_statistical_mechanics/6zMsNYYG">Nonequilibrium statistical mechanics</a> – Physics of many interacting particlesPages displaying short descriptions of redirect targets</li></ul> <h2 id="notes">Notes</h2> <h2 id="sources">Sources</h2> <ul><li>Baek, Yongjoo; Solon, Alexandre P.; Xu, Xinpeng; Nikola, Nikolai; Kafri, Yariv (2018-01-31). 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"Active Particles in Complex and Crowded Environments". <i>Reviews of Modern Physics</i>. 88 (4). American Physical Society (APS). <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2Frevmodphys.88.045006">10.1103/revmodphys.88.045006</a>. <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/11693%2F36533">11693/36533</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0034-6861">0034-6861</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:14940249">14940249</a>.</li> <li><a href="/facts/Michael_Cates/EzDKIDty">Cates, Michael E.</a>; Tailleur, Julien (2015-03-01). "Motility-Induced Phase Separation". <i>Annual Review of Condensed Matter Physics</i>. 6 (1). 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Springer Science and Business Media LLC: 2301–2317. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1512.07567">1512.07567</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1140%2Fepjst%2Fe2016-60062-0">10.1140/epjst/e2016-60062-0</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1951-6355">1951-6355</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:255387461">255387461</a>.</li> <li>Romanczuk, P.; Bär, M.; Ebeling, W.; Lindner, B.; Schimansky-Geier, L. (2012). "Active Brownian particles". <i>The European Physical Journal Special Topics</i>. 202 (1). Springer Science and Business Media LLC: 1–162. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1202.2442">1202.2442</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1140%2Fepjst%2Fe2012-01529-y">10.1140/epjst/e2012-01529-y</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1951-6355">1951-6355</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:255389128">255389128</a>.</li> <li>Santra, Ion; Basu, Urna; Sabhapandit, Sanjib (13 July 2021). "Active Brownian motion with directional reversals". <i>Physical Review E</i>. 104 (1): L012601. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/2101.11327">2101.11327</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2FPhysRevE.104.L012601">10.1103/PhysRevE.104.L012601</a>. <a href="/facts/EISSN_(identifier)/DPAflDvU">eISSN</a> <a href="https://search.worldcat.org/issn/2470-0053">2470-0053</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/2470-0045">2470-0045</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/34412243">34412243</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:231718971">231718971</a>.</li> <li>Shaebani, M. Reza; Wysocki, Adam; Winkler, Roland G.; Gompper, Gerhard; Rieger, Heiko (2020-03-10). "Computational models for active matter". <i>Nature Reviews Physics</i>. 2 (4). Springer Science and Business Media LLC: 181–199. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1910.02528">1910.02528</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1038%2Fs42254-020-0152-1">10.1038/s42254-020-0152-1</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/2522-5820">2522-5820</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:203836019">203836019</a>.</li> <li>Zöttl, Andreas; Stark, Holger (2016-05-11). "Emergent behavior in active colloids". <i>Journal of Physics: Condensed Matter</i>. 28 (25). IOP Publishing: 253001. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1601.06643">1601.06643</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0953-8984%2F28%2F25%2F253001">10.1088/0953-8984/28/25/253001</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0953-8984">0953-8984</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:3948148">3948148</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Howse, Jonathan R.; Jones, Richard A. L.; Ryan, Anthony J.; Gough, Tim; Vafabakhsh, Reza; Golestanian, Ramin (2007-07-27). "Self-Motile Colloidal Particles: From Directed Propulsion to Random Walk". Physical Review Letters. 99 (4): 048102. arXiv:0706.4406. doi:10.1103/PhysRevLett.99.048102. <a href="https://link.aps.org/doi/10.1103/PhysRevLett.99.048102" target="_blank">https://link.aps.org/doi/10.1103/PhysRevLett.99.048102</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Marchetti et al. 2016. - 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. 21. Elsevier BV: 34–43. arXiv:1510.00425. doi:10.1016/j.cocis.2016.01.003. ISSN 1359-0294. S2CID 97138568. <a href="https://arxiv.org/abs/1510.00425" target="_blank">https://arxiv.org/abs/1510.00425</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Zöttl & Stark 2016. - Zöttl, Andreas; Stark, Holger (2016-05-11). "Emergent behavior in active colloids". Journal of Physics: Condensed Matter. 28 (25). IOP Publishing: 253001. arXiv:1601.06643. doi:10.1088/0953-8984/28/25/253001. ISSN 0953-8984. S2CID 3948148. <a href="https://arxiv.org/abs/1601.06643" target="_blank">https://arxiv.org/abs/1601.06643</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Zöttl & Stark 2016. - Zöttl, Andreas; Stark, Holger (2016-05-11). "Emergent behavior in active colloids". Journal of Physics: Condensed Matter. 28 (25). IOP Publishing: 253001. arXiv:1601.06643. doi:10.1088/0953-8984/28/25/253001. ISSN 0953-8984. S2CID 3948148. <a href="https://arxiv.org/abs/1601.06643" target="_blank">https://arxiv.org/abs/1601.06643</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Peruani 2016. - Peruani, Fernando (2016). "Active Brownian rods". The European Physical Journal Special Topics. 225 (11–12). Springer Science and Business Media LLC: 2301–2317. arXiv:1512.07567. doi:10.1140/epjst/e2016-60062-0. ISSN 1951-6355. S2CID 255387461. <a href="https://arxiv.org/abs/1512.07567" target="_blank">https://arxiv.org/abs/1512.07567</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Bechinger et al. 2016. - 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. 88 (4). American Physical Society (APS). doi:10.1103/revmodphys.88.045006. hdl:11693/36533. ISSN 0034-6861. S2CID 14940249. <a href="https://doi.org/10.1103%2Frevmodphys.88.045006" target="_blank">https://doi.org/10.1103%2Frevmodphys.88.045006</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Howse, Jonathan R.; Jones, Richard A. L.; Ryan, Anthony J.; Gough, Tim; Vafabakhsh, Reza; Golestanian, Ramin (2007-07-27). "Self-Motile Colloidal Particles: From Directed Propulsion to Random Walk". 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