594:
517:
388:
to an external load they move with a constant velocity proportional to the applied force, just as objects in viscous media. Swirlons attract each other and coalesce forming a larger, joint swirlon. The coalescence is an extremely slow, decelerating process, resulting in a rarified state of immobile quasi-particles. In addition to the swirlonic state, gaseous, liquid and solid states were observed, depending on the inter-particle and self-driving forces. In contrast to molecular systems, liquid and gaseous states of self-propelled particles do not coexist.
606:
leader, yet the flock know exactly how to land in a unified way. The need for the group to land overrides deviating intentions by individual birds. The particle model found that the collective shift to landing depends on perturbations that apply to the individual birds, such as where the birds are in the flock. It is behaviour that can be compared with the way that sand avalanches, if it is piled up, before the point at which symmetric and carefully placed grains would avalanche, because the fluctuations become increasingly non-linear.
609:"Our main motivation was to better understand something which is puzzling and out there in nature, especially in cases involving the stopping or starting of a collective behavioural pattern in a group of people or animals ... We propose a simple model for a system whose members have the tendency to follow the others both in space and in their state of mind concerning a decision about stopping an activity. This is a very general model, which can be applied to similar situations." The model could also be applied to a swarm of unmanned
585:, the average density of marching bands is 50 locusts/m (50 million locusts/km), with a typical range from 20 to 120 locusts/m. The research findings discussed above demonstrate the dynamic instability that is present at the lower locust densities typical in the field, where marching groups randomly switch direction without any external perturbation. Understanding this phenomenon, together with the switch to fully coordinated marching at higher densities, is essential if the swarming of desert locusts is to be controlled.
3973:
22:
562:
451:
285:
2139:
549:
densities, they start falling into line and marching together, punctuated by abrupt but coordinated changes in direction. However, when densities reached a critical value at about 74 locusts/m, the locusts ceased making rapid and spontaneous changes in direction, and instead marched steadily in the same direction for the full eight hours of the experiment.
232:. Natural systems which have inspired the study and design of these particles include walking, swimming or flying animals. Other biological systems include bacteria, cells, algae and other micro-organisms. Generally, self-propelled particles often refer to artificial systems such as robots or specifically designed particles such as swimming
537:. If food is short they can gather together and start occupying neighbouring areas, recruiting more locusts. Eventually they can become a marching army extending over many kilometres. This can be the prelude to the development of the vast flying adult locust swarms which devastate vegetation on a continental scale.
466:
Simulations demonstrate that a suitable "nearest neighbour rule" eventually results in all the particles swarming together or moving in the same direction. This emerges, even though there is no centralised coordination, and even though the neighbours for each particle constantly change over time (see
370:
Symmetry breaking is a necessary condition for SPPs, as there must be a preferential direction for moving. However, the symmetry breaking may not come solely from the structure itself but from its interaction with electromagnetic fields, in particular when taken into account retardation effects. This
256:
To understand the ubiquity of such phenomena, physicists have developed a number of self-propelled particles models. These models predict that self-propelled particles share certain properties at the group level, regardless of the type of animals (or artificial particles) in the swarm. It has become
387:
reported a hitherto unrecognised state of self-propelled particles — which they called a "swirlonic state". The swirlonic state consists of "swirlons", formed by groups of self-propelled particles orbiting a common centre of mass. These quasi-particles demonstrate a surprising behaviour: In response
311:
allows, by properly tuning the environment (typically the surrounding solution), for the motion of the Janus particle. For instance, the two sides of the Janus particle can induce a local gradient of, temperature, electric field, or concentration of chemical species. This induces motion of the Janus
605:
In 2010, Bhattacharya and Vicsek used an SPP model to analyse what is happening here. As a paradigm, they considered how flying birds arrive at a collective decision to make a sudden and synchronised change to land. The birds, such as the starlings in the image on the right, have no decision-making
601:
Swarming animals, such as ants, bees, fish and birds, are often observed suddenly switching from one state to another. For example, birds abruptly switch from a flying state to a landing state. Or fish switch from schooling in one direction to schooling in another direction. Such state switches can
275:
transform ATP energy into directional motion. Recent work has shown that enzyme molecules will also propel themselves. Further, it has been shown that they will preferentially move towards a region of higher substrate concentration, a phenomenon that has been developed into a purification technique
270:
Most animals can be seen as SPP: they find energy in their food and exhibit various locomotion strategies, from flying to crawling. The most prominent examples of collective behaviours in these systems are fish schools, birds flocks, sheep herds, human crowds. At a smaller scale, cells and bacteria
361:
Walking grains are a typical realization of dry SPP: The grains are milli-metric disks sitting on a vertically vibrating plate, which serves as the source of energy and momentum. The disks have two different contacts ("feet") with the plate, a hard needle-like foot in the front and a large soft
544:
of a group increases, an abrupt transition occurs from individuals moving in relatively disordered and independent ways within the group to the group moving as a highly aligned whole. Thus, in the case of young desert locusts, a trigger point should occur which turns disorganised and dispersed
548:
In 2006, a group of researchers examined how this model held up in the laboratory. Locusts were placed in a circular arena, and their movements were tracked with computer software. At low densities, below 18 locusts per square metre, the locusts mill about in a disordered way. At intermediate
414:
imply directed motion is either the spontaneous formation of clusters or the separation in a gas-like and a liquid-like phase, an unexpected phenomenon when the SPP have purely repulsive interaction. This phase separation has been called
Motility Induced Phase Separation (MIPS).
484:
Body interactions: the particles can be considered as points (no body interaction) like in the Vicsek model. Alternatively, one can include an interaction potential, either attractive or repulsive. This potential can be isotropic or not to describe spherical or elongated
252:
Self-propelled particles interact with each other, which can lead to the emergence of collective behaviours. These collective behaviours mimic the self-organization observed with the flocking of birds, the swarming of bugs, the formation of sheep herds, etc.
332:
and
Mallouk. In a solution of hydrogen peroxide, this "nanomotor" would exhibit a catalytic oxidation-reduction reaction, thereby inducing a fluid flow along the surface through self-diffusiophoresis. A similar system used a copper-platinum rod in a bromine
336:
Another Janus SPP was developed by coating half of a polystyrene bead with platinum. These were used to direct the motion of catalytic motors when they were close to a solid surface. These systems were able to move the active colloids using geometric
276:
to isolate live enzymes. Additionally, microparticles or vesicles can become self-propelled when they are functionalized with enzymes. The catalytic reactions of the enzymes direct the particles or vesicles based on corresponding substrate gradients.
324:. Because the Janus particles consume energy from their environment (catalysis of chemical reactions, light absorption, etc.), the resulting motion constitutes an irreversible process and the particles are out of equilibrium.
499:
Self-propelled particles can also be modeled using on-lattice models, which offer the advantage of being simple and efficient to simulate, and in some cases, may be easier to analyze mathematically. On-lattice models such as
438:. In that case the SPP are point particles, which move with a constant speed. and adopt (at each time increment) the average direction of motion of the other particles in their local neighborhood up to some added noise.
362:
rubber foot in the back. When shaken, the disks move in a preferential direction defined by the polar (head-tail) symmetry of the contacts. This together with the vibrational noise result in a persistent random walk.
488:
Body orientation: for those particles with a body-fixed axis, one can include additional degrees of freedom to describe the orientation of the body. The coupling of this body axis with the velocity is an additional
344:
was tethered to the silica half of the particle and in solution of monomer would drive a catalytic polymerization. The resulting concentration gradient across the surface would propel the motor in solution.
496:
One can also include effective influences of the surrounding; for instance the nominal velocity of the SPP can be set to depend on the local density, in order to take into account crowding effects.
3577:
1824:
Meredith, Caleb H.; Castonguay, Alexander C.; Chiu, Yu-Jen; Brooks, Allan M.; Moerman, Pepijn G.; Torab, Peter; Wong, Pak Kin; Sen, Ayusman; Velegol, Darrell; Zarzar, Lauren D. (2 February 2022).
244:, observed in biological systems, e.g. bacteria quorum sensing and ant pheromone detection, and in synthetic systems, e.g. enzyme molecule chemotaxis and enzyme powered hard and soft particles.
504:
models have been used to study physical aspects of self-propelled particle systems (such as phase transitions and pattern-forming potential) as well as specific questions related to real
602:
occur with astonishing speed and synchronicity, as though all the members in the group made a unanimous decision at the same moment. Phenomena like these have long puzzled researchers.
197:
3581:
545:
locusts into a coordinated marching army. When the critical population density is reached, the insects should start marching together in a stable way and in the same direction.
717:
296:
There is a distinction between wet and dry systems. In the first case the particles "swim" in a surrounding fluid; in the second case the particles "walk" on a substrate.
1333:
Somasundar A, Ghosh S, Mohajerani F, Massenburg LN, Yang T, Cremer PS, et al. (December 2019). "Positive and negative chemotaxis of enzyme-coated liposome motors".
2560:
Tektonidis M, Hatzikirou H, Chauvière A, Simon M, Schaller K, Deutsch A (October 2011). "Identification of intrinsic in vitro cellular mechanisms for glioma invasion".
492:
Aligning interaction rules: in the spirit of the Vicsek model, neighboring particles align their velocities. Another possibility is that they align their orientations.
371:
can be used for the phototactic motion of even highly symmetrical nanoparticles. In 2021, it was experimentally shown that completely symmetric particles (spherical
1411:
3665:
3409:
582:
2868:
Baglietto G, Albano EV (November 2009). "Nature of the order-disorder transition in the Vicsek model for the collective motion of self-propelled particles".
1550:
Paxton WF, Kistler KC, Olmeda CC, Sen A, St Angelo SK, Cao Y, et al. (October 2004). "Catalytic nanomotors: autonomous movement of striped nanorods".
407:. In that case all particles move in the same direction. On top of that, spatial structures can emerge such as bands, vortices, asters, moving clusters.
474:
to detailed and specialized models aiming at describing specific systems and situations. Among the important ingredients in these models, one can list
4315:
1239:
Dey KK, Zhao X, Tansi BM, Méndez-Ortiz WJ, Córdova-Figueroa UM, Golestanian R, Sen A (December 2015). "Micromotors
Powered by Enzyme Catalysis".
2171:
Zhang, Jianhua; Laskar, Abhrajit; Song, Jiaqi; Shklyaev, Oleg E.; Mou, Fangzhi; Guan, Jianguo; Balazs, Anna C.; Sen, Ayusman (10 January 2023).
2799:
271:
can also be treated as SPP. These biological systems can propel themselves based on the presence of chemoattractants. At even smaller scale,
347:
Another example of an artificial SPP are platinum spinner microparticles that have controllable rotations based on their shape and symmetry.
3206:
Bertin E, Droz M, Grégoire G (2009). "Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis".
965:
Bertin E, Droz M, Grégoire G (2009). "Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis".
204:
3609:
Aditi Simha R, Ramaswamy S (July 2002). "Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles".
2820:
2229:
Vicsek T, Czirók A, Ben-Jacob E, Cohen I, Shochet O (August 1995). "Novel type of phase transition in a system of self-driven particles".
112:
107:
613:, to initiate the desired motion in a crowd of people, or to interpreting group patterns when stock market shares are bought or sold.
764:
3359:
Czirók A, Barabási AL, Vicsek T (1999). "Collective motion of self-propelled particles: Kinetic phase transition in one dimension".
2155:
597:
Flocks of birds can abruptly change their direction in unison, and then, just as suddenly, make a unanimous group decision to land
4004:
2911:
Chowdhury D (2006). "Collective effects in intra-cellular molecular motor transport: coordination, cooperation and competition".
2434:
2762:
Somfai E, Czirok A, Vicsek T (1994). "Power-law distribution of landslides in an experiment on the erosion of a granular pile".
4184:
662:
1021:
3923:
3677:
3599:
3421:
1405:
792:
Howse, Jonathan R.; Jones, Richard A. L.; Ryan, Anthony J.; Gough, Tim; Vafabakhsh, Reza; Golestanian, Ramin (27 July 2007).
454:
77:
2173:"Light-Powered, Fuel-Free Oscillation, Migration, and Reversible Manipulation of Multiple Cargo Types by Micromotor Swarms"
2046:
Fränzl M, Muiños-Landin S, Holubec V, Cichos F (February 2021). "Fully
Steerable Symmetric Thermoplasmonic Microswimmers".
1120:
Sengupta S, Dey KK, Muddana HS, Tabouillot T, Ibele ME, Butler PJ, Sen A (January 2013). "Enzyme molecules as nanomotors".
1932:
Che, Shengping; Zhang, Jianhua; Mou, Fangzhi; Guo, Xia; Kauffman, Joshua E.; Sen, Ayusman; Guan, Jianguo (January 2022).
1385:
907:
152:
62:
2396:
Jadbabaie A, Lin J, Morse AS (2003). "Coordination of groups of mobile autonomous agents using nearest neighbor rules".
3933:
3820:
2317:
593:
435:
328:
The first example of an artificial SPP on the nano or micron scale was a gold-platinum bimetallic nanorod developed by
240:
and walking grains. In the case of directed propulsion, which is driven by a chemical gradient, this is referred to as
167:
4126:
2629:
565:
1282:
Ghosh S, Mohajerani F, Son S, Velegol D, Butler PJ, Sen A (September 2019). "Motility of Enzyme-Powered
Vesicles".
117:
4341:
4023:
3088:
Bickel T, Majee A, Würger A (July 2013). "Flow pattern in the vicinity of self-propelling hot Janus particles".
572:
When the density of locusts reaches a critical point, they march steadily together without direction reversals.
1997:
Zhang, Jianhua; Mou, Fangzhi; Tang, Shaowen; Kauffman, Joshua E.; Sen, Ayusman; Guan, Jianguo (1 March 2022).
3792:
72:
3255:
Ihle T (October 2013). "Invasion-wave-induced first-order phase transition in systems of active particles".
1999:"Photochemical micromotor of eccentric core in isotropic hollow shell exhibiting multimodal motion behavior"
403:
The prominent and most spectacular emergent large scale behaviour observed in assemblies of SPP is directed
4351:
3894:
3845:
67:
37:
400:
generally includes the formation of self-assembled structures, such as clusters and organized assemblies.
4141:
4094:
3472:
Levine H, Rappel WJ, Cohen I (January 2001). "Self-organization in systems of self-propelled particles".
307:
are colloidal particles with two different sides, having different physical or chemical properties. This
190:
2292:
Proceedings of the 14th annual conference on
Computer graphics and interactive techniques - SIGGRAPH '87
3850:
3529:
1677:
Pavlick RA, Sengupta S, McFadden T, Zhang H, Sen A (September 2011). "A polymerization-powered motor".
87:
1825:
849:
Agudo-Canalejo, Jaime; Adeleke-Larodo, Tunrayo; Illien, Pierre; Golestanian, Ramin (16 October 2018).
4028:
3999:
3899:
3837:
2643:
Huepe C, Aldana M (April 2004). "Intermittency and clustering in a system of self-driven particles".
2607:. Grasshopper and locust: a handbook of general acridology. Vol. II. Cambridge University Press.
1585:
Liu R, Sen A (December 2011). "Autonomous nanomotor based on copper-platinum segmented nanobattery".
257:
a challenge in theoretical physics to find minimal statistical models that capture these behaviours.
2410:
2300:
3310:
Czirók A, Stanley HE, Vicsek T (1997). "Spontaneously ordered motion of self-propelled particles".
508:
systems (for example, identifying the underlying biological processes involved in tumor invasion).
471:
516:
4260:
4045:
3361:
2966:
Helbing D, Farkas I, Vicsek T (September 2000). "Simulating dynamical features of escape panic".
668:
610:
384:
157:
122:
4280:
3940:
3813:
2851:
2405:
2295:
229:
52:
3027:
Helbing D, Keltsch J, Molnár P (July 1997). "Modelling the evolution of human trail systems".
4089:
3994:
375:
in this case) experience a net thermophoretic force when illuminated from a given direction.
2622:
4300:
4265:
4103:
4050:
3957:
3700:
3628:
3548:
3491:
3444:
3380:
3331:
3312:
3274:
3227:
3208:
3176:
3107:
3046:
2985:
2932:
2877:
2773:
2764:
2710:
2652:
2569:
2362:
2248:
2102:
1945:
1890:
1871:
Deseigne J, Dauchot O, Chaté H (August 2010). "Collective motion of vibrated polar disks".
1733:
1633:
1516:
1456:
1342:
1291:
1248:
1038:
986:
967:
924:
815:
732:
340:
Another example of a Janus SPP is an organometallic motor using a gold-silica microsphere.
162:
2341:
Czirók A, Vicsek T (2006). "Collective behavior of interacting self-propelled particles".
8:
4336:
4310:
4285:
4275:
3865:
3433:"Self-propelled particles with soft-core interactions: patterns, stability, and collapse"
3239:
1434:
1397:
998:
793:
341:
4113:
3704:
3632:
3552:
3495:
3448:
3384:
3335:
3278:
3231:
3180:
3111:
3050:
2989:
2936:
2881:
2777:
2722:
2714:
2656:
2573:
2470:
Philosophical
Transactions of the Royal Society of London. Series B, Biological Sciences
2366:
2252:
2106:
1949:
1894:
1737:
1637:
1520:
1460:
1346:
1295:
1252:
1042:
990:
928:
850:
819:
736:
4290:
4151:
4146:
3726:
3652:
3618:
3564:
3538:
3515:
3481:
3396:
3370:
3347:
3321:
3298:
3264:
3243:
3217:
3166:
3131:
3097:
3070:
3036:
3009:
2975:
2948:
2922:
2843:
2748:
2726:
2700:
2542:
2515:"First steps on asynchronous lattice-gas models with an application to a swarming rule"
2490:
2465:
2378:
2352:
2323:
2272:
2238:
2208:
2123:
2090:
2071:
2028:
1974:
1933:
1914:
1880:
1853:
1806:
1754:
1722:"Shape-directed rotation of homogeneous micromotors via catalytic self-electrophoresis"
1721:
1702:
1654:
1621:
1506:
1446:
1366:
1315:
1180:
1155:
Zhao X, Gentile K, Mohajerani F, Sen A (October 2018). "Powering Motion with
Enzymes".
1097:
1072:
1002:
976:
862:
805:
756:
716:
Buhl J, Sumpter DJ, Couzin ID, Hale JJ, Despland E, Miller ER, Simpson SJ (June 2006).
541:
481:: in the absence of interaction, the SPP speed converges to a prescribed constant value
45:
26:
2785:
2744:
2442:
2374:
4346:
4255:
4174:
4161:
4121:
3962:
3806:
3718:
3673:
3644:
3595:
3568:
3507:
3460:
3417:
3343:
3290:
3194:
3123:
3062:
3001:
2893:
2668:
2585:
2534:
2495:
2313:
2264:
2212:
2200:
2192:
2128:
2075:
2063:
2032:
2020:
1979:
1961:
1906:
1857:
1845:
1810:
1798:
1759:
1694:
1659:
1602:
1567:
1532:
1472:
1401:
1370:
1358:
1319:
1307:
1264:
1221:
1172:
1137:
1102:
915:
888:
880:
831:
748:
404:
397:
308:
3761:
3730:
3519:
3400:
3351:
3302:
3247:
3135:
2952:
2847:
2730:
2382:
2276:
1918:
1620:
Das S, Garg A, Campbell AI, Howse J, Sen A, Velegol D, et al. (December 2015).
1528:
1184:
1006:
4356:
4136:
4084:
4067:
3989:
3981:
3708:
3656:
3636:
3587:
3556:
3527:
Mehandia V, Nott PR (2008). "The collective dynamics of self-propelled particles".
3499:
3452:
3388:
3339:
3282:
3235:
3184:
3115:
3074:
3054:
2993:
2940:
2885:
2835:
2781:
2718:
2660:
2577:
2546:
2526:
2485:
2477:
2415:
2370:
2305:
2256:
2184:
2172:
2118:
2110:
2055:
2010:
1969:
1953:
1902:
1898:
1837:
1790:
1778:
1749:
1741:
1706:
1686:
1649:
1641:
1594:
1559:
1524:
1464:
1435:"Propulsion of a Molecular Machine by Asymmetric Distribution of Reaction Products"
1393:
1350:
1299:
1256:
1211:
1198:
Dey KK, Das S, Poyton MF, Sengupta S, Butler PJ, Cremer PS, Sen A (December 2014).
1164:
1129:
1092:
1084:
1054:
1046:
994:
932:
872:
823:
740:
321:
225:
3640:
3456:
3013:
2691:
Bhattacharya K, Vicsek T (2010). "Collective decision making in cohesive flocks".
2664:
2464:
Nava-Sedeño JM, Voß-Böhme A, Hatzikirou H, Deutsch A, Peruani F (September 2020).
2327:
1468:
827:
760:
3952:
3829:
2804:
2290:
Reynolds CW (1987). "Flocks, herds and schools: A distributed behavioral model".
1720:
Brooks AM, Tasinkevych M, Sabrina S, Velegol D, Sen A, Bishop KJ (January 2019).
1303:
1260:
1168:
876:
677:
646:
630:
534:
478:
350:
Another example is biphasic Janus oil droplets which shows self propelled motion.
317:
304:
289:
272:
233:
3739:
2944:
2260:
1777:
Unruh, Angus; Brooks, Allan M.; Aranson, Igor S.; Sen, Ayusman (28 April 2023).
1050:
424:
4295:
4229:
4206:
4040:
4035:
4018:
4011:
3904:
3503:
3432:
3392:
3286:
3189:
3154:
3119:
2889:
2581:
2143:
2114:
2015:
1998:
1841:
1745:
1494:
622:
313:
177:
134:
3972:
3787:
3560:
2745:"Self-Propelled Particle System Improves Understanding Of Behavioral Patterns"
2530:
2466:"Modelling collective cell motion: are on- and off-lattice models equivalent?"
1354:
940:
936:
4330:
4169:
3911:
3882:
3860:
3591:
2538:
2196:
2024:
1965:
1849:
1802:
1536:
1433:
Golestanian, Ramin; Liverpool, Tanniemola B.; Ajdari, Armand (10 June 2005).
884:
848:
626:
530:
505:
2514:
2419:
2188:
2059:
1957:
1794:
744:
21:
4250:
4224:
4214:
4191:
4072:
3757:
3722:
3648:
3511:
3464:
3294:
3198:
3127:
3005:
2897:
2672:
2589:
2499:
2481:
2268:
2204:
2132:
2067:
1983:
1910:
1763:
1698:
1690:
1663:
1606:
1571:
1476:
1362:
1311:
1268:
1225:
1176:
1141:
1106:
892:
835:
752:
689:
683:
665: – Tendency of self-propelled particles to aggregate and form clusters
372:
3066:
794:"Self-Motile Colloidal Particles: From Directed Propulsion to Random Walk"
470:
Since then a number of models have been proposed, ranging from the simple
4270:
4219:
4060:
3916:
3623:
3486:
3375:
3326:
3041:
2980:
2839:
2357:
2243:
1511:
1451:
642:
329:
2927:
2309:
2156:
Swirlonic Super
Particles: Physicists Baffled by a Novel State of Matter
4077:
4055:
3753:
1779:"Programming Motion of Platinum Microparticles: From Linear to Orbital"
1645:
241:
237:
29:
occur in swarms independent of the type of animal that is in the swarm.
1598:
1563:
1216:
1199:
1133:
1088:
1071:
Muddana HS, Sengupta S, Mallouk TE, Sen A, Butler PJ (February 2010).
4305:
3945:
2997:
2913:
2463:
2343:
2089:
Brilliantov NV, Abutuqayqah H, Tyukin IY, Matveev SA (October 2020).
1029:
1022:"Minimal mechanisms for school formation in self-propelled particles"
354:
300:
3713:
3688:
4234:
4196:
3928:
3870:
2088:
1332:
867:
650:
634:
501:
97:
3762:"Refining self-propelled particle models for collective behaviour"
3543:
3269:
3222:
3171:
3155:"Kinetic theory of flocking: derivation of hydrodynamic equations"
3102:
3058:
2834:(5). Finnish Zoological and Botanical Publishing Board.: 415–428.
2705:
2559:
1885:
981:
810:
3855:
3430:
2142:
Material was copied from this source, which is available under a
2045:
284:
3889:
3741:
On the dynamics and evolution of self-propelled particle models
2821:"Key behavioural factors in a self-organised fish school model"
1719:
127:
102:
3431:
D'Orsogna MR, Chuang YL, Bertozzi AL, Chayes LS (March 2006).
4131:
2138:
638:
431:
288:
An example of a SPP: a gold-platinum nanorod which undergoes
228:, which convert energy from the environment into directed or
1823:
1676:
1119:
3877:
3798:
540:
One of the key predictions of the SPP model is that as the
2228:
1493:
Golestanian, R.; Liverpool, T. B.; Ajdari, A. (May 2007).
1492:
1432:
692: – Mathematical model used to describe active matter.
621:
SPP models have been applied in many other areas, such as
578:
This confirmed the behaviour predicted by the SPP models.
3410:"Flocking: collective motion of self-propelled particles"
1281:
1070:
1019:
1934:"Light-Programmable Assemblies of Isotropic Micromotors"
1238:
1154:
292:
in hydrogen peroxide due to self-electrophoretic forces.
3747:(MSc thesis). Somerville College, University of Oxford.
2170:
673:
Pages displaying short descriptions of redirect targets
583:
Food and
Agriculture Organization of the United Nations
2144:
Creative Commons Attribution 4.0 International License
1549:
1073:"Substrate catalysis enhances single-enzyme diffusion"
3578:"The wonderful world of active many-particle systems"
2818:
2605:
Behaviour, ecology, biogeography, population dynamics
1776:
715:
467:
the interactive simulation in the box on the right).
3751:
3608:
3358:
3026:
1996:
1870:
1619:
1197:
851:"Enhanced Diffusion and Chemotaxis at the Nanoscale"
511:
3309:
2965:
791:
410:Another class of large scale behaviour, which does
312:particle along the gradient through, respectively,
4316:Task allocation and partitioning of social insects
3205:
2761:
2690:
2512:
1826:"Chemical design of self-propelled Janus droplets"
964:
3471:
3087:
2623:"Desert locust guidelines: Biology and behaviour"
2395:
686: – Microscopic object able to traverse fluid
4328:
2819:Gautrais J, Jost C, Theraulaz G (October 2008).
2737:
2636:
2513:Bouré O, Fatès N, Chevrier V (1 December 2013).
905:
3081:
2867:
2686:
2684:
2682:
2620:
391:
2792:
2765:Journal of Physics A: Mathematical and General
2441:. University of Colorado. 2005. Archived from
1931:
423:The modeling of SPP was introduced in 1995 by
3814:
3416:. Oxford University Press. pp. 177–209.
2389:
2224:
2222:
1495:"Designing phoretic micro- and nano-swimmers"
1020:Li YX, Lukeman R, Edelstein-Keshet L (2007).
680: – Type of nanoparticle or microparticle
198:
3689:"Statistical physics: Closing in on evaders"
3526:
3407:
2679:
2340:
718:"From disorder to order in marching locusts"
711:
709:
707:
705:
353:Several other examples are described in the
2642:
1622:"Boundaries can steer active Janus spheres"
1383:
958:
303:, are the prototypical example of wet SPP.
3821:
3807:
2219:
2039:
205:
191:
3712:
3622:
3542:
3485:
3374:
3325:
3268:
3221:
3188:
3170:
3101:
3040:
2979:
2926:
2910:
2704:
2489:
2409:
2356:
2299:
2242:
2122:
2014:
1973:
1884:
1753:
1653:
1510:
1450:
1215:
1096:
980:
866:
809:
702:
224:are terms used by physicists to describe
2289:
1587:Journal of the American Chemical Society
1552:Journal of the American Chemical Society
1122:Journal of the American Chemical Society
1077:Journal of the American Chemical Society
592:
515:
418:
283:
20:
3663:
3575:
1013:
4329:
3769:Canadian Applied Mathematics Quarterly
3686:
2616:
2614:
2602:
2398:IEEE Transactions on Automatic Control
1584:
663:Clustering of self-propelled particles
3924:Patterns of self-organization in ants
3802:
3737:
3145:
2800:"Bird flock decision-making revealed"
2424:convergence proofs for the SPP model.
1488:
1486:
1414:from the original on 23 February 2024
279:
265:
3254:
3152:
908:"Hydrodynamics and phases of flocks"
787:
785:
365:
3414:Fluctuations and scaling in biology
2611:
1200:"Chemotactic separation of enzymes"
906:Toner J, Tu Y, Ramaswamy S (2005).
524:
299:Active colloidal particles, dubbed
153:Distributed artificial intelligence
63:Agent-based computational economics
13:
3934:symmetry breaking of escaping ants
3586:. Vol. 41. pp. 357–368.
2091:"Swirlonic state of active matter"
1483:
1398:10.1093/oso/9780192858313.003.0008
168:Multi-agent reinforcement learning
14:
4368:
3781:
1783:ACS Applied Engineering Materials
782:
616:
512:Some applications to real systems
3971:
2294:. Vol. 21. pp. 25–34.
2137:
671: – Type of bacterial motion
588:
560:
455:SPP model interactive simulation
449:
3583:Advances in Solid State Physics
3020:
2959:
2904:
2861:
2812:
2755:
2621:Symmons PM, Cressman K (2001).
2596:
2553:
2506:
2457:
2427:
2334:
2283:
2164:
2149:
2082:
1990:
1925:
1864:
1817:
1770:
1713:
1670:
1613:
1578:
1543:
1426:
1377:
1326:
1275:
1232:
1191:
581:In the field, according to the
3672:. Princeton University Press.
3240:10.1088/1751-8113/42/44/445001
2562:Journal of Theoretical Biology
1903:10.1103/PhysRevLett.105.098001
1148:
1113:
1064:
1030:Physica D: Nonlinear Phenomena
999:10.1088/1751-8113/42/44/445001
899:
842:
641:in urban green spaces. SPP in
383:In 2020, researchers from the
1:
3641:10.1103/PhysRevLett.89.058101
3457:10.1103/PhysRevLett.96.104302
2723:10.1088/1367-2630/12/9/093019
2665:10.1103/PhysRevLett.92.168701
2375:10.1016/S0378-4371(00)00013-3
1469:10.1103/PhysRevLett.94.220801
1157:Accounts of Chemical Research
855:Accounts of Chemical Research
828:10.1103/PhysRevLett.99.048102
696:
113:Platforms for software agents
78:Agent-based modeling software
73:Agent-based social simulation
3895:Mixed-species foraging flock
3846:Agent-based model in biology
3828:
3666:"Chapter 5: Moving together"
2435:"Self driven particle model"
1304:10.1021/acs.nanolett.9b01830
1261:10.1021/acs.nanolett.5b03935
1169:10.1021/acs.accounts.8b00286
877:10.1021/acs.accounts.8b00280
434:model introduced in 1986 by
392:Typical collective behaviour
68:Agent-based model in biology
7:
4142:Particle swarm optimization
3408:Czirók A, Vicsek T (2001).
2945:10.1016/j.physa.2006.05.005
2786:10.1088/0305-4470/27/20/001
2261:10.1103/PhysRevLett.75.1226
1384:Golestanian, Ramin (2022).
1051:10.1016/j.physd.2007.10.009
656:
649:, are often modeled by the
637:and the evolution of human
378:
260:
247:
220:(SPP), also referred to as
10:
4373:
3851:Collective animal behavior
3670:Collective Animal Behavior
3530:Journal of Fluid Mechanics
3504:10.1103/PhysRevE.63.017101
3393:10.1103/PhysRevLett.82.209
3344:10.1088/0305-4470/30/5/009
3287:10.1103/PhysRevE.88.040303
3190:10.1103/PhysRevE.83.030901
3120:10.1103/PhysRevE.88.012301
2890:10.1103/PhysRevE.80.050103
2582:10.1016/j.jtbi.2011.07.012
2115:10.1038/s41598-020-73824-4
2016:10.1016/j.apmt.2022.101371
1842:10.1016/j.matt.2021.12.014
1746:10.1038/s41467-019-08423-7
533:are solitary and wingless
88:Agent-oriented programming
25:SPP models predict robust
4243:
4205:
4160:
4112:
3980:
3969:
3836:
3561:10.1017/S0022112007009184
2828:Annales Zoologici Fennici
2531:10.1007/s11047-013-9389-2
1529:10.1088/1367-2630/9/5/126
1355:10.1038/s41565-019-0578-8
937:10.1016/j.aop.2005.04.011
559:
554:
448:
443:
430:as a special case of the
4180:Self-propelled particles
3592:10.1007/3-540-44946-9_29
1386:"Phoretic Active Matter"
472:active Brownian particle
218:Self-propelled particles
173:Self-propelled particles
16:Type of autonomous agent
4261:Collective intelligence
4127:Ant colony optimization
3788:Swarming desert locusts
3611:Physical Review Letters
3437:Physical Review Letters
3362:Physical Review Letters
2645:Physical Review Letters
2439:Interactive simulations
2420:10.1109/TAC.2003.812781
2231:Physical Review Letters
2189:10.1021/acsnano.2c07266
2060:10.1021/acsnano.0c10598
2003:Applied Materials Today
1873:Physical Review Letters
1795:10.1021/acsaenm.2c00249
1439:Physical Review Letters
798:Physical Review Letters
745:10.1126/science.1125142
669:Run-and-tumble particle
385:University of Leicester
236:, bimetallic nanorods,
158:Multi-agent pathfinding
4342:Complex systems theory
4281:Microbial intelligence
3941:Shoaling and schooling
3687:Vicsek T (July 2010).
2693:New Journal of Physics
2482:10.1098/rstb.2019.0378
1691:10.1002/anie.201103565
1499:New Journal of Physics
598:
521:
293:
230:persistent random walk
222:self-driven particles,
53:Multi-agent simulation
30:
3412:. In Vicsek T (ed.).
3153:Ihle T (March 2011).
1958:10.34133/2022/9816562
1726:Nature Communications
1626:Nature Communications
1335:Nature Nanotechnology
633:, the development of
596:
519:
419:Examples of modelling
287:
24:
4301:Spatial organization
4266:Decentralised system
4104:Sea turtle migration
3958:Swarming (honey bee)
3313:Journal of Physics A
3209:Journal of Physics A
2840:10.5735/086.045.0505
2808:. 14 September 2010.
2751:. 18 September 2010.
1392:. pp. 230–293.
968:Journal of Physics A
770:on 29 September 2011
163:Multi-agent planning
4352:Multi-agent systems
4276:Group size measures
3838:Biological swarming
3705:2010Natur.466...43V
3664:Sumpter DJ (2010).
3633:2002PhRvL..89e8101A
3553:2008JFM...595..239M
3496:2000PhRvE..63a7101L
3449:2006PhRvL..96j4302D
3385:1999PhRvL..82..209C
3336:1997JPhA...30.1375C
3279:2013PhRvE..88d0303I
3232:2009JPhA...42R5001B
3181:2011PhRvE..83c0901I
3112:2013PhRvE..88a2301B
3051:1997Natur.388...47H
2990:2000Natur.407..487H
2937:2006PhyA..372...84C
2882:2009PhRvE..80e0103B
2857:on 12 January 2011.
2778:1994JPhA...27L.757S
2715:2010NJPh...12i3019B
2657:2004PhRvL..92p8701H
2574:2011JThBi.287..131T
2367:2000PhyA..281...17C
2310:10.1145/37401.37406
2253:1995PhRvL..75.1226V
2161:, 11 February 2021.
2107:2020NatSR..1016783B
1950:2022Resea202216562C
1895:2010PhRvL.105i8001D
1738:2019NatCo..10..495B
1638:2015NatCo...6.8999D
1521:2007NJPh....9..126G
1461:2005PhRvL..94v0801G
1347:2019NatNa..14.1129S
1296:2019NanoL..19.6019G
1253:2015NanoL..15.8311D
1043:2008PhyD..237..699L
991:2009JPhA...42R5001B
929:2005AnPhy.318..170T
820:2007PhRvL..99d8102H
737:2006Sci...312.1402B
46:Multi-agent systems
27:emergent behaviours
4291:Predator satiation
4152:Swarm (simulation)
4147:Swarm intelligence
4122:Agent-based models
3953:Swarming behaviour
3790:– Video clip from
3576:Helbing D (2001).
3480:(1 Pt 2): 017101.
3165:(3 Pt 1): 030901.
3146:Further references
2876:(5 Pt 1): 050103.
2749:Medical News Today
2603:Uvarov BP (1977).
2476:(1807): 20190378.
2445:on 14 October 2012
2095:Scientific Reports
1646:10.1038/ncomms9999
1060:on 1 October 2011.
599:
542:population density
522:
294:
280:Artificial systems
266:Biological systems
31:
4324:
4323:
4311:Military swarming
4256:Animal navigation
4175:Collective motion
4162:Collective motion
4029:reverse migration
3963:Swarming motility
3738:Yates CA (2007).
3679:978-0-691-12963-1
3601:978-3-540-42000-2
3474:Physical Review E
3423:978-0-19-850790-1
3257:Physical Review E
3159:Physical Review E
3090:Physical Review E
2870:Physical Review E
2772:(20): L757–L763.
2747:(Press release).
2519:Natural Computing
1679:Angewandte Chemie
1599:10.1021/ja2082735
1564:10.1021/ja047697z
1407:978-0-19-285831-3
1341:(12): 1129–1134.
1217:10.1021/nn504418u
1163:(10): 2373–2381.
1134:10.1021/ja3091615
1089:10.1021/ja908773a
916:Annals of Physics
861:(10): 2365–2372.
576:
575:
464:
463:
405:collective motion
398:collective motion
366:Symmetry breaking
309:symmetry breaking
226:autonomous agents
215:
214:
4364:
4137:Crowd simulation
4114:Swarm algorithms
4085:Insect migration
3990:Animal migration
3982:Animal migration
3975:
3900:Mobbing behavior
3823:
3816:
3809:
3800:
3799:
3776:
3766:
3748:
3746:
3734:
3716:
3683:
3660:
3626:
3624:cond-mat/0108301
3605:
3572:
3546:
3523:
3489:
3487:cond-mat/0006477
3468:
3427:
3404:
3378:
3376:cond-mat/9712154
3355:
3329:
3327:cond-mat/0611741
3320:(5): 1375–1385.
3306:
3272:
3251:
3225:
3202:
3192:
3174:
3140:
3139:
3105:
3085:
3079:
3078:
3044:
3042:cond-mat/9805158
3024:
3018:
3017:
2998:10.1038/35035023
2983:
2981:cond-mat/0009448
2974:(6803): 487–90.
2963:
2957:
2956:
2930:
2908:
2902:
2901:
2865:
2859:
2858:
2856:
2850:. Archived from
2825:
2816:
2810:
2809:
2796:
2790:
2789:
2759:
2753:
2752:
2741:
2735:
2734:
2708:
2688:
2677:
2676:
2640:
2634:
2633:
2627:
2618:
2609:
2608:
2600:
2594:
2593:
2557:
2551:
2550:
2510:
2504:
2503:
2493:
2461:
2455:
2454:
2452:
2450:
2431:
2425:
2423:
2413:
2393:
2387:
2386:
2360:
2358:cond-mat/0611742
2338:
2332:
2331:
2303:
2287:
2281:
2280:
2246:
2244:cond-mat/0611743
2237:(6): 1226–1229.
2226:
2217:
2216:
2168:
2162:
2153:
2147:
2141:
2136:
2126:
2086:
2080:
2079:
2054:(2): 3434–3440.
2043:
2037:
2036:
2018:
1994:
1988:
1987:
1977:
1929:
1923:
1922:
1888:
1868:
1862:
1861:
1821:
1815:
1814:
1789:(4): 1126–1133.
1774:
1768:
1767:
1757:
1717:
1711:
1710:
1674:
1668:
1667:
1657:
1617:
1611:
1610:
1582:
1576:
1575:
1558:(41): 13424–31.
1547:
1541:
1540:
1514:
1512:cond-mat/0701168
1490:
1481:
1480:
1454:
1452:cond-mat/0701169
1430:
1424:
1423:
1421:
1419:
1390:academic.oup.com
1381:
1375:
1374:
1330:
1324:
1323:
1290:(9): 6019–6026.
1279:
1273:
1272:
1236:
1230:
1229:
1219:
1195:
1189:
1188:
1152:
1146:
1145:
1117:
1111:
1110:
1100:
1068:
1062:
1061:
1059:
1053:. Archived from
1026:
1017:
1011:
1010:
984:
962:
956:
955:
953:
951:
945:
939:. Archived from
923:(170): 170–244.
912:
903:
897:
896:
870:
846:
840:
839:
813:
789:
780:
779:
777:
775:
769:
763:. Archived from
731:(5778): 1402–6.
722:
713:
674:
631:molecular motors
569:– sped up 6-fold
566:Marching locusts
564:
563:
552:
551:
525:Marching locusts
453:
452:
441:
440:
342:Grubb's catalyst
322:diffusiophoresis
273:molecular motors
207:
200:
193:
33:
32:
4372:
4371:
4367:
4366:
4365:
4363:
4362:
4361:
4327:
4326:
4325:
4320:
4239:
4201:
4156:
4108:
3976:
3967:
3832:
3827:
3784:
3779:
3764:
3744:
3714:10.1038/466043a
3680:
3602:
3424:
3148:
3143:
3086:
3082:
3035:(6637): 47–50.
3025:
3021:
2964:
2960:
2928:physics/0605053
2909:
2905:
2866:
2862:
2854:
2823:
2817:
2813:
2805:Himalayan Times
2798:
2797:
2793:
2760:
2756:
2743:
2742:
2738:
2689:
2680:
2641:
2637:
2625:
2619:
2612:
2601:
2597:
2558:
2554:
2511:
2507:
2462:
2458:
2448:
2446:
2433:
2432:
2428:
2411:10.1.1.128.5326
2404:(6): 988–1001.
2394:
2390:
2339:
2335:
2320:
2301:10.1.1.103.7187
2288:
2284:
2227:
2220:
2169:
2165:
2154:
2150:
2087:
2083:
2044:
2040:
1995:
1991:
1930:
1926:
1869:
1865:
1822:
1818:
1775:
1771:
1718:
1714:
1675:
1671:
1618:
1614:
1593:(50): 20064–7.
1583:
1579:
1548:
1544:
1491:
1484:
1431:
1427:
1417:
1415:
1408:
1382:
1378:
1331:
1327:
1280:
1276:
1237:
1233:
1210:(12): 11941–9.
1196:
1192:
1153:
1149:
1118:
1114:
1069:
1065:
1057:
1024:
1018:
1014:
963:
959:
949:
947:
946:on 18 July 2011
943:
910:
904:
900:
847:
843:
790:
783:
773:
771:
767:
720:
714:
703:
699:
678:Janus particles
672:
659:
647:Janus particles
635:human stampedes
619:
591:
571:
570:
561:
555:External videos
527:
514:
479:Self-propulsion
460:
457:
450:
444:External videos
421:
394:
381:
368:
357:-specific page.
318:electrophoresis
305:Janus particles
290:self-propulsion
282:
268:
263:
250:
211:
182:
139:
82:
17:
12:
11:
5:
4370:
4360:
4359:
4354:
4349:
4344:
4339:
4322:
4321:
4319:
4318:
4313:
4308:
4303:
4298:
4296:Quorum sensing
4293:
4288:
4283:
4278:
4273:
4268:
4263:
4258:
4253:
4247:
4245:
4244:Related topics
4241:
4240:
4238:
4237:
4232:
4230:Swarm robotics
4227:
4222:
4217:
4211:
4209:
4207:Swarm robotics
4203:
4202:
4200:
4199:
4194:
4189:
4188:
4187:
4177:
4172:
4166:
4164:
4158:
4157:
4155:
4154:
4149:
4144:
4139:
4134:
4129:
4124:
4118:
4116:
4110:
4109:
4107:
4106:
4101:
4100:
4099:
4098:
4097:
4082:
4081:
4080:
4075:
4065:
4064:
4063:
4058:
4053:
4048:
4041:Fish migration
4038:
4036:Cell migration
4033:
4032:
4031:
4026:
4019:Bird migration
4016:
4015:
4014:
4012:coded wire tag
4009:
4008:
4007:
3997:
3986:
3984:
3978:
3977:
3970:
3968:
3966:
3965:
3960:
3955:
3950:
3949:
3948:
3938:
3937:
3936:
3931:
3921:
3920:
3919:
3909:
3908:
3907:
3905:feeding frenzy
3897:
3892:
3887:
3886:
3885:
3875:
3874:
3873:
3868:
3858:
3853:
3848:
3842:
3840:
3834:
3833:
3826:
3825:
3818:
3811:
3803:
3797:
3796:
3783:
3782:External links
3780:
3778:
3777:
3749:
3735:
3699:(7302): 43–4.
3684:
3678:
3661:
3606:
3600:
3573:
3524:
3469:
3443:(10): 104302.
3428:
3422:
3405:
3369:(1): 209–212.
3356:
3307:
3252:
3216:(44): 445001.
3203:
3149:
3147:
3144:
3142:
3141:
3080:
3019:
2958:
2903:
2860:
2811:
2791:
2754:
2736:
2678:
2651:(16): 168701.
2635:
2610:
2595:
2552:
2525:(4): 551–560.
2505:
2456:
2426:
2388:
2333:
2319:978-0897912273
2318:
2282:
2218:
2183:(1): 251–262.
2163:
2148:
2081:
2038:
1989:
1924:
1863:
1836:(2): 616–633.
1816:
1769:
1712:
1685:(40): 9374–7.
1669:
1612:
1577:
1542:
1482:
1445:(22): 220801.
1425:
1406:
1376:
1325:
1274:
1247:(12): 8311–5.
1231:
1190:
1147:
1128:(4): 1406–14.
1112:
1063:
1037:(5): 699–720.
1012:
975:(44): 445001.
957:
898:
841:
781:
700:
698:
695:
694:
693:
687:
681:
675:
666:
658:
655:
627:robotic swarms
623:schooling fish
618:
617:Other examples
615:
590:
587:
574:
573:
568:
557:
556:
531:desert locusts
526:
523:
513:
510:
494:
493:
490:
486:
482:
462:
461:
458:
446:
445:
420:
417:
393:
390:
380:
377:
367:
364:
359:
358:
351:
348:
345:
338:
334:
314:thermophoresis
281:
278:
267:
264:
262:
259:
249:
246:
234:Janus colloids
213:
212:
210:
209:
202:
195:
187:
184:
183:
181:
180:
178:Swarm robotics
175:
170:
165:
160:
155:
149:
146:
145:
141:
140:
138:
137:
135:Software agent
132:
131:
130:
125:
120:
110:
105:
100:
94:
91:
90:
84:
83:
81:
80:
75:
70:
65:
59:
56:
55:
49:
48:
42:
41:
15:
9:
6:
4:
3:
2:
4369:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4338:
4335:
4334:
4332:
4317:
4314:
4312:
4309:
4307:
4304:
4302:
4299:
4297:
4294:
4292:
4289:
4287:
4284:
4282:
4279:
4277:
4274:
4272:
4269:
4267:
4264:
4262:
4259:
4257:
4254:
4252:
4249:
4248:
4246:
4242:
4236:
4233:
4231:
4228:
4226:
4223:
4221:
4218:
4216:
4213:
4212:
4210:
4208:
4204:
4198:
4195:
4193:
4190:
4186:
4183:
4182:
4181:
4178:
4176:
4173:
4171:
4170:Active matter
4168:
4167:
4165:
4163:
4159:
4153:
4150:
4148:
4145:
4143:
4140:
4138:
4135:
4133:
4130:
4128:
4125:
4123:
4120:
4119:
4117:
4115:
4111:
4105:
4102:
4096:
4093:
4092:
4091:
4088:
4087:
4086:
4083:
4079:
4076:
4074:
4071:
4070:
4069:
4066:
4062:
4059:
4057:
4054:
4052:
4049:
4047:
4046:diel vertical
4044:
4043:
4042:
4039:
4037:
4034:
4030:
4027:
4025:
4022:
4021:
4020:
4017:
4013:
4010:
4006:
4003:
4002:
4001:
3998:
3996:
3993:
3992:
3991:
3988:
3987:
3985:
3983:
3979:
3974:
3964:
3961:
3959:
3956:
3954:
3951:
3947:
3944:
3943:
3942:
3939:
3935:
3932:
3930:
3927:
3926:
3925:
3922:
3918:
3915:
3914:
3913:
3910:
3906:
3903:
3902:
3901:
3898:
3896:
3893:
3891:
3888:
3884:
3883:herd behavior
3881:
3880:
3879:
3876:
3872:
3869:
3867:
3864:
3863:
3862:
3859:
3857:
3854:
3852:
3849:
3847:
3844:
3843:
3841:
3839:
3835:
3831:
3824:
3819:
3817:
3812:
3810:
3805:
3804:
3801:
3795:
3794:
3789:
3786:
3785:
3774:
3770:
3763:
3760:(Fall 2010).
3759:
3755:
3750:
3743:
3742:
3736:
3732:
3728:
3724:
3720:
3715:
3710:
3706:
3702:
3698:
3694:
3690:
3685:
3681:
3675:
3671:
3667:
3662:
3658:
3654:
3650:
3646:
3642:
3638:
3634:
3630:
3625:
3620:
3617:(5): 058101.
3616:
3612:
3607:
3603:
3597:
3593:
3589:
3585:
3584:
3579:
3574:
3570:
3566:
3562:
3558:
3554:
3550:
3545:
3540:
3536:
3532:
3531:
3525:
3521:
3517:
3513:
3509:
3505:
3501:
3497:
3493:
3488:
3483:
3479:
3475:
3470:
3466:
3462:
3458:
3454:
3450:
3446:
3442:
3438:
3434:
3429:
3425:
3419:
3415:
3411:
3406:
3402:
3398:
3394:
3390:
3386:
3382:
3377:
3372:
3368:
3364:
3363:
3357:
3353:
3349:
3345:
3341:
3337:
3333:
3328:
3323:
3319:
3315:
3314:
3308:
3304:
3300:
3296:
3292:
3288:
3284:
3280:
3276:
3271:
3266:
3263:(4): 040303.
3262:
3258:
3253:
3249:
3245:
3241:
3237:
3233:
3229:
3224:
3219:
3215:
3211:
3210:
3204:
3200:
3196:
3191:
3186:
3182:
3178:
3173:
3168:
3164:
3160:
3156:
3151:
3150:
3137:
3133:
3129:
3125:
3121:
3117:
3113:
3109:
3104:
3099:
3096:(1): 012301.
3095:
3091:
3084:
3076:
3072:
3068:
3064:
3060:
3059:10.1038/40353
3056:
3052:
3048:
3043:
3038:
3034:
3030:
3023:
3015:
3011:
3007:
3003:
2999:
2995:
2991:
2987:
2982:
2977:
2973:
2969:
2962:
2954:
2950:
2946:
2942:
2938:
2934:
2929:
2924:
2920:
2916:
2915:
2907:
2899:
2895:
2891:
2887:
2883:
2879:
2875:
2871:
2864:
2853:
2849:
2845:
2841:
2837:
2833:
2829:
2822:
2815:
2807:
2806:
2801:
2795:
2787:
2783:
2779:
2775:
2771:
2767:
2766:
2758:
2750:
2746:
2740:
2732:
2728:
2724:
2720:
2716:
2712:
2707:
2702:
2699:(9): 093019.
2698:
2694:
2687:
2685:
2683:
2674:
2670:
2666:
2662:
2658:
2654:
2650:
2646:
2639:
2631:
2624:
2617:
2615:
2606:
2599:
2591:
2587:
2583:
2579:
2575:
2571:
2567:
2563:
2556:
2548:
2544:
2540:
2536:
2532:
2528:
2524:
2520:
2516:
2509:
2501:
2497:
2492:
2487:
2483:
2479:
2475:
2471:
2467:
2460:
2444:
2440:
2436:
2430:
2421:
2417:
2412:
2407:
2403:
2399:
2392:
2384:
2380:
2376:
2372:
2368:
2364:
2359:
2354:
2350:
2346:
2345:
2337:
2329:
2325:
2321:
2315:
2311:
2307:
2302:
2297:
2293:
2286:
2278:
2274:
2270:
2266:
2262:
2258:
2254:
2250:
2245:
2240:
2236:
2232:
2225:
2223:
2214:
2210:
2206:
2202:
2198:
2194:
2190:
2186:
2182:
2178:
2174:
2167:
2160:
2157:
2152:
2145:
2140:
2134:
2130:
2125:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2092:
2085:
2077:
2073:
2069:
2065:
2061:
2057:
2053:
2049:
2042:
2034:
2030:
2026:
2022:
2017:
2012:
2008:
2004:
2000:
1993:
1985:
1981:
1976:
1971:
1967:
1963:
1959:
1955:
1951:
1947:
1943:
1939:
1935:
1928:
1920:
1916:
1912:
1908:
1904:
1900:
1896:
1892:
1887:
1882:
1879:(9): 098001.
1878:
1874:
1867:
1859:
1855:
1851:
1847:
1843:
1839:
1835:
1831:
1827:
1820:
1812:
1808:
1804:
1800:
1796:
1792:
1788:
1784:
1780:
1773:
1765:
1761:
1756:
1751:
1747:
1743:
1739:
1735:
1731:
1727:
1723:
1716:
1708:
1704:
1700:
1696:
1692:
1688:
1684:
1680:
1673:
1665:
1661:
1656:
1651:
1647:
1643:
1639:
1635:
1631:
1627:
1623:
1616:
1608:
1604:
1600:
1596:
1592:
1588:
1581:
1573:
1569:
1565:
1561:
1557:
1553:
1546:
1538:
1534:
1530:
1526:
1522:
1518:
1513:
1508:
1504:
1500:
1496:
1489:
1487:
1478:
1474:
1470:
1466:
1462:
1458:
1453:
1448:
1444:
1440:
1436:
1429:
1413:
1409:
1403:
1399:
1395:
1391:
1387:
1380:
1372:
1368:
1364:
1360:
1356:
1352:
1348:
1344:
1340:
1336:
1329:
1321:
1317:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1285:
1278:
1270:
1266:
1262:
1258:
1254:
1250:
1246:
1242:
1235:
1227:
1223:
1218:
1213:
1209:
1205:
1201:
1194:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1151:
1143:
1139:
1135:
1131:
1127:
1123:
1116:
1108:
1104:
1099:
1094:
1090:
1086:
1083:(7): 2110–1.
1082:
1078:
1074:
1067:
1056:
1052:
1048:
1044:
1040:
1036:
1032:
1031:
1023:
1016:
1008:
1004:
1000:
996:
992:
988:
983:
978:
974:
970:
969:
961:
942:
938:
934:
930:
926:
922:
918:
917:
909:
902:
894:
890:
886:
882:
878:
874:
869:
864:
860:
856:
852:
845:
837:
833:
829:
825:
821:
817:
812:
807:
804:(4): 048102.
803:
799:
795:
788:
786:
766:
762:
758:
754:
750:
746:
742:
738:
734:
730:
726:
719:
712:
710:
708:
706:
701:
691:
688:
685:
682:
679:
676:
670:
667:
664:
661:
660:
654:
652:
648:
644:
640:
636:
632:
628:
624:
614:
612:
607:
603:
595:
589:Bird landings
586:
584:
579:
567:
558:
553:
550:
546:
543:
538:
536:
532:
518:
509:
507:
506:active matter
503:
497:
491:
487:
483:
480:
477:
476:
475:
473:
468:
456:
447:
442:
439:
437:
433:
429:
426:
416:
413:
408:
406:
401:
399:
389:
386:
376:
374:
373:microswimmers
363:
356:
352:
349:
346:
343:
339:
335:
331:
327:
326:
325:
323:
319:
315:
310:
306:
302:
297:
291:
286:
277:
274:
258:
254:
245:
243:
239:
235:
231:
227:
223:
219:
208:
203:
201:
196:
194:
189:
188:
186:
185:
179:
176:
174:
171:
169:
166:
164:
161:
159:
156:
154:
151:
150:
148:
147:
143:
142:
136:
133:
129:
126:
124:
121:
119:
116:
115:
114:
111:
109:
106:
104:
101:
99:
96:
95:
93:
92:
89:
86:
85:
79:
76:
74:
71:
69:
66:
64:
61:
60:
58:
57:
54:
51:
50:
47:
44:
43:
39:
35:
34:
28:
23:
19:
4251:Allee effect
4225:Nanorobotics
4215:Ant robotics
4192:Vicsek model
4179:
3793:Planet Earth
3791:
3772:
3768:
3740:
3696:
3692:
3669:
3614:
3610:
3582:
3534:
3528:
3477:
3473:
3440:
3436:
3413:
3366:
3360:
3317:
3311:
3260:
3256:
3213:
3207:
3162:
3158:
3093:
3089:
3083:
3032:
3028:
3022:
2971:
2967:
2961:
2921:(1): 84–95.
2918:
2912:
2906:
2873:
2869:
2863:
2852:the original
2831:
2827:
2814:
2803:
2794:
2769:
2763:
2757:
2739:
2696:
2692:
2648:
2644:
2638:
2604:
2598:
2565:
2561:
2555:
2522:
2518:
2508:
2473:
2469:
2459:
2447:. Retrieved
2443:the original
2438:
2429:
2401:
2397:
2391:
2351:(1): 17–29.
2348:
2342:
2336:
2291:
2285:
2234:
2230:
2180:
2176:
2166:
2159:SciTechDaily
2158:
2151:
2101:(1): 16783.
2098:
2094:
2084:
2051:
2047:
2041:
2006:
2002:
1992:
1941:
1937:
1927:
1876:
1872:
1866:
1833:
1829:
1819:
1786:
1782:
1772:
1729:
1725:
1715:
1682:
1678:
1672:
1629:
1625:
1615:
1590:
1586:
1580:
1555:
1551:
1545:
1502:
1498:
1442:
1438:
1428:
1416:. Retrieved
1389:
1379:
1338:
1334:
1328:
1287:
1284:Nano Letters
1283:
1277:
1244:
1241:Nano Letters
1240:
1234:
1207:
1203:
1193:
1160:
1156:
1150:
1125:
1121:
1115:
1080:
1076:
1066:
1055:the original
1034:
1028:
1015:
972:
966:
960:
948:. Retrieved
941:the original
920:
914:
901:
858:
854:
844:
801:
797:
772:. Retrieved
765:the original
728:
724:
690:Vicsek model
684:Microswimmer
620:
608:
604:
600:
580:
577:
547:
539:
528:
520:Locust nymph
498:
495:
469:
465:
459:– needs Java
427:
425:Tamás Vicsek
422:
411:
409:
402:
395:
382:
369:
360:
337:constraints.
298:
295:
269:
255:
251:
221:
217:
216:
172:
18:
4271:Eusociality
4220:Microbotics
4090:butterflies
4061:sardine run
3995:altitudinal
3917:pack hunter
3756:, Erban R,
3537:: 239–264.
643:Stokes flow
4337:Biophysics
4331:Categories
4185:clustering
4078:philopatry
4056:salmon run
4051:Lessepsian
3752:Yates CA,
2568:: 131–47.
2009:: 101371.
1732:(1): 495.
1505:(5): 126.
868:2104.02398
697:References
645:, such as
485:particles.
301:nanomotors
242:chemotaxis
238:nanomotors
4306:Stigmergy
4286:Mutualism
3946:bait ball
3569:119610757
3544:0707.1436
3270:1304.0149
3223:0907.4688
3172:1006.1825
3103:1401.7311
2914:Physica A
2706:1007.4453
2539:1572-9796
2406:CiteSeerX
2344:Physica A
2296:CiteSeerX
2213:253257444
2197:1936-0851
2076:231874669
2033:246188941
2025:2352-9407
1966:2639-5274
1886:1004.1499
1858:246203036
1850:2590-2385
1811:257849071
1803:2771-9545
1537:1367-2630
1371:208168622
1320:201095514
982:0907.4688
885:0001-4842
811:0706.4406
355:nanomotor
333:solution.
4347:Ethology
4235:Symbrion
4197:BIO-LGCA
4000:tracking
3929:ant mill
3871:sort sol
3866:flocking
3830:Swarming
3758:Maini PK
3754:Baker RE
3731:12682238
3723:20596010
3649:12144468
3520:19509007
3512:11304390
3465:16605738
3401:16881098
3352:16154002
3303:14951536
3295:24229097
3248:17686543
3199:21517447
3136:36558271
3128:23944457
3006:11028994
2953:14822256
2898:20364937
2848:16940460
2731:32835905
2673:15169268
2628:. Rome:
2590:21816160
2500:32713300
2449:10 April
2383:14211016
2277:15918052
2269:10060237
2205:36321936
2177:ACS Nano
2133:33033334
2068:33556235
2048:ACS Nano
1984:35928302
1938:Research
1919:40192049
1911:20868196
1764:30700714
1699:21948434
1664:26627125
1632:: 8999.
1607:21961523
1572:15479099
1477:16090376
1418:23 April
1412:Archived
1363:31740796
1312:31429577
1269:26587897
1226:25243599
1204:ACS Nano
1185:52845451
1177:30256612
1142:23308365
1107:20108965
1007:17686543
893:30240187
836:17678409
753:16741126
657:See also
651:squirmer
502:BIO-LGCA
436:Reynolds
396:Typical
379:Swirlons
261:Examples
248:Overview
98:Auto-GPT
38:a series
36:Part of
4357:Zoology
4095:monarch
4024:flyways
4005:history
3856:Droving
3701:Bibcode
3657:3845736
3629:Bibcode
3549:Bibcode
3492:Bibcode
3445:Bibcode
3381:Bibcode
3332:Bibcode
3275:Bibcode
3228:Bibcode
3177:Bibcode
3108:Bibcode
3075:4364517
3067:9214501
3047:Bibcode
2986:Bibcode
2933:Bibcode
2878:Bibcode
2774:Bibcode
2711:Bibcode
2653:Bibcode
2570:Bibcode
2547:6794567
2491:7423376
2363:Bibcode
2249:Bibcode
2124:7546729
2103:Bibcode
1975:9297725
1946:Bibcode
1891:Bibcode
1755:6353883
1734:Bibcode
1707:6325323
1655:4686856
1634:Bibcode
1517:Bibcode
1457:Bibcode
1343:Bibcode
1292:Bibcode
1249:Bibcode
1098:2832858
1039:Bibcode
987:Bibcode
950:7 April
925:Bibcode
816:Bibcode
774:7 April
733:Bibcode
725:Science
653:model.
489:option.
144:Related
103:Botnets
4068:Homing
3890:Locust
3729:
3721:
3693:Nature
3676:
3655:
3647:
3598:
3567:
3518:
3510:
3463:
3420:
3399:
3350:
3301:
3293:
3246:
3197:
3134:
3126:
3073:
3065:
3029:Nature
3014:310346
3012:
3004:
2968:Nature
2951:
2896:
2846:
2729:
2671:
2588:
2545:
2537:
2498:
2488:
2408:
2381:
2328:546350
2326:
2316:
2298:
2275:
2267:
2211:
2203:
2195:
2131:
2121:
2074:
2066:
2031:
2023:
1982:
1972:
1964:
1917:
1909:
1856:
1848:
1830:Matter
1809:
1801:
1762:
1752:
1705:
1697:
1662:
1652:
1605:
1570:
1535:
1475:
1404:
1369:
1361:
1318:
1310:
1267:
1224:
1183:
1175:
1140:
1105:
1095:
1005:
891:
883:
834:
761:359329
759:
751:
639:trails
611:drones
535:nymphs
529:Young
428:et al.
128:GORITE
4132:Boids
4073:natal
3861:Flock
3765:(PDF)
3745:(PDF)
3727:S2CID
3653:S2CID
3619:arXiv
3565:S2CID
3539:arXiv
3516:S2CID
3482:arXiv
3397:S2CID
3371:arXiv
3348:S2CID
3322:arXiv
3299:S2CID
3265:arXiv
3244:S2CID
3218:arXiv
3167:arXiv
3132:S2CID
3098:arXiv
3071:S2CID
3037:arXiv
3010:S2CID
2976:arXiv
2949:S2CID
2923:arXiv
2855:(PDF)
2844:S2CID
2824:(PDF)
2727:S2CID
2701:arXiv
2626:(PDF)
2543:S2CID
2379:S2CID
2353:arXiv
2324:S2CID
2273:S2CID
2239:arXiv
2209:S2CID
2072:S2CID
2029:S2CID
1915:S2CID
1881:arXiv
1854:S2CID
1807:S2CID
1703:S2CID
1507:arXiv
1447:arXiv
1367:S2CID
1316:S2CID
1181:S2CID
1058:(PDF)
1025:(PDF)
1003:S2CID
977:arXiv
944:(PDF)
911:(PDF)
863:arXiv
806:arXiv
768:(PDF)
757:S2CID
721:(PDF)
432:Boids
3912:Pack
3878:Herd
3775:(3).
3719:PMID
3674:ISBN
3645:PMID
3596:ISBN
3508:PMID
3461:PMID
3418:ISBN
3291:PMID
3195:PMID
3124:PMID
3063:PMID
3002:PMID
2894:PMID
2669:PMID
2586:PMID
2535:ISSN
2496:PMID
2451:2011
2314:ISBN
2265:PMID
2201:PMID
2193:ISSN
2129:PMID
2064:PMID
2021:ISSN
1980:PMID
1962:ISSN
1942:2022
1907:PMID
1846:ISSN
1799:ISSN
1760:PMID
1695:PMID
1660:PMID
1603:PMID
1568:PMID
1533:ISSN
1473:PMID
1420:2024
1402:ISBN
1359:PMID
1308:PMID
1265:PMID
1222:PMID
1173:PMID
1138:PMID
1103:PMID
952:2011
889:PMID
881:ISSN
832:PMID
776:2011
749:PMID
123:JACK
118:JADE
108:FIPA
3709:doi
3697:466
3637:doi
3588:doi
3557:doi
3535:595
3500:doi
3453:doi
3389:doi
3340:doi
3283:doi
3236:doi
3185:doi
3116:doi
3055:doi
3033:388
2994:doi
2972:407
2941:doi
2919:372
2886:doi
2836:doi
2782:doi
2719:doi
2661:doi
2630:FAO
2578:doi
2566:287
2527:doi
2486:PMC
2478:doi
2474:375
2416:doi
2371:doi
2349:281
2306:doi
2257:doi
2185:doi
2119:PMC
2111:doi
2056:doi
2011:doi
1970:PMC
1954:doi
1899:doi
1877:105
1838:doi
1791:doi
1750:PMC
1742:doi
1687:doi
1650:PMC
1642:doi
1595:doi
1591:133
1560:doi
1556:126
1525:doi
1465:doi
1394:doi
1351:doi
1300:doi
1257:doi
1212:doi
1165:doi
1130:doi
1126:135
1093:PMC
1085:doi
1081:132
1047:doi
1035:237
995:doi
933:doi
921:318
873:doi
824:doi
741:doi
729:312
412:not
330:Sen
320:or
4333::
3773:18
3771:.
3767:.
3725:.
3717:.
3707:.
3695:.
3691:.
3668:.
3651:.
3643:.
3635:.
3627:.
3615:89
3613:.
3594:.
3580:.
3563:.
3555:.
3547:.
3533:.
3514:.
3506:.
3498:.
3490:.
3478:63
3476:.
3459:.
3451:.
3441:96
3439:.
3435:.
3395:.
3387:.
3379:.
3367:82
3365:.
3346:.
3338:.
3330:.
3318:30
3316:.
3297:.
3289:.
3281:.
3273:.
3261:88
3259:.
3242:.
3234:.
3226:.
3214:42
3212:.
3193:.
3183:.
3175:.
3163:83
3161:.
3157:.
3130:.
3122:.
3114:.
3106:.
3094:88
3092:.
3069:.
3061:.
3053:.
3045:.
3031:.
3008:.
3000:.
2992:.
2984:.
2970:.
2947:.
2939:.
2931:.
2917:.
2892:.
2884:.
2874:80
2872:.
2842:.
2832:45
2830:.
2826:.
2802:.
2780:.
2770:27
2768:.
2725:.
2717:.
2709:.
2697:12
2695:.
2681:^
2667:.
2659:.
2649:92
2647:.
2613:^
2584:.
2576:.
2564:.
2541:.
2533:.
2523:12
2521:.
2517:.
2494:.
2484:.
2472:.
2468:.
2437:.
2414:.
2402:48
2400:.
2377:.
2369:.
2361:.
2347:.
2322:.
2312:.
2304:.
2271:.
2263:.
2255:.
2247:.
2235:75
2233:.
2221:^
2207:.
2199:.
2191:.
2181:17
2179:.
2175:.
2127:.
2117:.
2109:.
2099:10
2097:.
2093:.
2070:.
2062:.
2052:15
2050:.
2027:.
2019:.
2007:26
2005:.
2001:.
1978:.
1968:.
1960:.
1952:.
1944:.
1940:.
1936:.
1913:.
1905:.
1897:.
1889:.
1875:.
1852:.
1844:.
1832:.
1828:.
1805:.
1797:.
1785:.
1781:.
1758:.
1748:.
1740:.
1730:10
1728:.
1724:.
1701:.
1693:.
1683:50
1681:.
1658:.
1648:.
1640:.
1628:.
1624:.
1601:.
1589:.
1566:.
1554:.
1531:.
1523:.
1515:.
1501:.
1497:.
1485:^
1471:.
1463:.
1455:.
1443:94
1441:.
1437:.
1410:.
1400:.
1388:.
1365:.
1357:.
1349:.
1339:14
1337:.
1314:.
1306:.
1298:.
1288:19
1286:.
1263:.
1255:.
1245:15
1243:.
1220:.
1206:.
1202:.
1179:.
1171:.
1161:51
1159:.
1136:.
1124:.
1101:.
1091:.
1079:.
1075:.
1045:.
1033:.
1027:.
1001:.
993:.
985:.
973:42
971:.
931:.
919:.
913:.
887:.
879:.
871:.
859:51
857:.
853:.
830:.
822:.
814:.
802:99
800:.
796:.
784:^
755:.
747:.
739:.
727:.
723:.
704:^
629:,
625:,
316:,
40:on
3822:e
3815:t
3808:v
3733:.
3711::
3703::
3682:.
3659:.
3639::
3631::
3621::
3604:.
3590::
3571:.
3559::
3551::
3541::
3522:.
3502::
3494::
3484::
3467:.
3455::
3447::
3426:.
3403:.
3391::
3383::
3373::
3354:.
3342::
3334::
3324::
3305:.
3285::
3277::
3267::
3250:.
3238::
3230::
3220::
3201:.
3187::
3179::
3169::
3138:.
3118::
3110::
3100::
3077:.
3057::
3049::
3039::
3016:.
2996::
2988::
2978::
2955:.
2943::
2935::
2925::
2900:.
2888::
2880::
2838::
2788:.
2784::
2776::
2733:.
2721::
2713::
2703::
2675:.
2663::
2655::
2632:.
2592:.
2580::
2572::
2549:.
2529::
2502:.
2480::
2453:.
2422:–
2418::
2385:.
2373::
2365::
2355::
2330:.
2308::
2279:.
2259::
2251::
2241::
2215:.
2187::
2146:.
2135:.
2113::
2105::
2078:.
2058::
2035:.
2013::
1986:.
1956::
1948::
1921:.
1901::
1893::
1883::
1860:.
1840::
1834:5
1813:.
1793::
1787:1
1766:.
1744::
1736::
1709:.
1689::
1666:.
1644::
1636::
1630:6
1609:.
1597::
1574:.
1562::
1539:.
1527::
1519::
1509::
1503:9
1479:.
1467::
1459::
1449::
1422:.
1396::
1373:.
1353::
1345::
1322:.
1302::
1294::
1271:.
1259::
1251::
1228:.
1214::
1208:8
1187:.
1167::
1144:.
1132::
1109:.
1087::
1049::
1041::
1009:.
997::
989::
979::
954:.
935::
927::
895:.
875::
865::
838:.
826::
818::
808::
778:.
743::
735::
206:e
199:t
192:v
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
