Self-propelled particles

SPP models predict robust emergent behaviours occur in swarms independent of the type of animal that is in the swarm.

Self-propelled particles (SPP), also referred to as self-driven particles, is a concept used by physicists to describe autonomous agents, which convert energy from the environment into directed or persistent motion. Everyday life examples of such agents are walking, swimming or flying animals. Other biological systems include bacteria, cells, algae and other micro-organisms. Generally, directed propulsion in biological systems is referred to as chemotaxis. One can also think of artificial systems such as robots or specifically designed particles such as swimming Janus colloids, nanomotors, walking grains, and others.

Overview

Self-propelled particles interact according to various social and physical rules, which can lead to the emergence of collective behaviors, such as flocking of birds, swarming of bugs, the formation of sheep herds, etc.

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 in the swarm.^[1] It has become a challenge in theoretical physics to find minimal statistical models that capture these behaviours.^[2][3][4]

Examples of Self-Propelled Particles

Biological systems

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 are fish schools, birds flocks, sheep herds, human crowds. At a smaller scale, cells and bacteria can also be treated as SPP. These biological systems can propel themselves based on the presence of chemoattractants. At even smaller scale, molecular motors transform ATP energy into directional motion. Recent work has shown that enzyme molecules will also propel themselves.^[5] Further, it has been shown that they will preferentially move towards a region of higher substrate concentration,^[6] a phenomenon that has been developed into a purification technique to isolate live enzymes.^[7]

Artificial systems

One usually distinguishes 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.

Active colloidal particles are the prototypical example of wet SPP: Janus particles are colloidal particles with two different sides, having different physical or chemical properties. This symmetry breaking 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 particle along the gradient through, respectively, thermophoresis, electrophoresis or diffusiophoresis. 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.

• The first example of an artificial SPP on the nano or micron scale was a gold-platinum bimetallic nanorod developed by Sen and Mallouk.^[8] 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 solution.^[9]
• Another example of a janus SPP is an organometallic motor using a gold-silica microsphere.^[10] Grubb's catalyst 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.

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 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.

Typical Collective Behavior

The prominent and most spectacular emergent large scale behavior observed in assemblies of SPP is directed collective motion. 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.

Another class of large scale behavior, which does not 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).

Examples of Modeling

The modeling of SPP was introduced in 1995 by Tamas Vicsek et al.^[11] as a special case of the Boids model introduced in 1986 by Reynolds.^[12] 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.^[13][14]

External video

SPP model interactive simulation[15] – needs Java

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 the interactive simulation in the box on the right).^[11]

Since then a number of models have been proposed, ranging from the simples so called Active Brownian Particle to highly elaborated and specialized models aiming at describing specific systems and situations. Among the important ingredients in these models, one can list

• Self-propulsion: in the absence of interaction, the SPP speed converges to a prescribed constant value
• 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 particles.
• 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 option.
• Aligning interaction rules: in the spirit of the Vicsek model, neighboring particles align their velocities. An other possibility is that they align their orientations.

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.

Some applications to real systems

Locust nymph

External video

Marching locusts – sped up 6-fold. When the density of locusts reaches a critical point, they march steadily together without direction reversals.

Marching locusts

Young desert locusts are solitary and wingless nymphs. 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.^[16] This can be the prelude to the development of the vast flying adult locust swarms which devastate vegetation on a continental scale.^[17]

One of the key predictions of the SPP model is that as the population density 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.^[18] Thus, in the case of young desert locusts, a trigger point should occur which turns disorganised and dispersed 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.

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 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/m², 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 (see video on the left). This confirmed the behaviour predicted by the SPP models.^[1]

In the field, according to the Food and Agriculture Organization of the United Nations, the average density of marching bands is 50 locusts/m² (50 million locusts/km²), with a typical range from 20 to 120 locusts/m².^[17]:29 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.^[1]

Bird landings

Flocks of birds can abruptly change their direction in unison, and then, just as suddenly, make a unanimous group decision to land^[19]

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 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.^[20]

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 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.^[19] 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.^[21]

"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."^[19] The model could also be applied to a swarm of unmanned drones, to initiating a desired motion in a crowd of people, or to interpreting group patterns when stock market shares are bought or sold.^[22]

Other examples

SPP models have been applied in many other areas, such as schooling fish,^[23] robotic swarms,^[24] molecular motors,^[25] the development of human stampedes^[26] and the evolution of human trails in urban green spaces.^[27] SPP in Stokes flow, such as Janus particles, are often modeled by the squirmer model,.^[28]

References

1. Buhl, J.; Sumpter, D. J. T.; Couzin, D.; Hale, J. J.; Despland, E.; Miller, E. R.; Simpson, S. J. (2006). "From disorder to order in marching locusts" (PDF). Science 312 (5778): 1402–1406. Bibcode:2006Sci...312.1402B. doi:10.1126/science.1125142. PMID 16741126.
2. Toner, J.; Tu, Y.; Ramaswamy, S. (2005). "Hydrodynamics and phases of flocks" (PDF). Annals of Physics 318 (170): 170. Bibcode:2005AnPhy.318..170T. doi:10.1016/j.aop.2005.04.011.
3. Bertin, E.; Droz, M.; Grégoire, G. (2009). "Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis". Journal of Physics A 42 (44): 445001. arXiv:0907.4688. Bibcode:2009JPhA...42R5001B. doi:10.1088/1751-8113/42/44/445001.
4. Li, Y. X.; Lukeman, R.; Edelstein-Keshet, L. (2007). "Minimal mechanisms for school formation in self-propelled particles" (PDF). Physica D: Nonlinear Phenomena 237 (5): 699–720. Bibcode:2008PhyD..237..699L. doi:10.1016/j.physd.2007.10.009.
5. Muddana, Hari S.; Sengupta, Samudra; Mallouk, Thomas E.; Sen, Ayusman; Butler, Peter J. (2010-02-24). "Substrate Catalysis Enhances Single Enzyme Diffusion". Journal of the American Chemical Society 132 (7): 2110–2111. doi:10.1021/ja908773a. ISSN 0002-7863. PMC 2832858. PMID 20108965.
6. Sengupta, Samudra; Dey, Krishna K.; Muddana, Hari S.; Tabouillot, Tristan; Ibele, Michael E.; Butler, Peter J.; Sen, Ayusman (2013-01-30). "Enzyme Molecules as Nanomotors". Journal of the American Chemical Society 135 (4): 1406–1414. doi:10.1021/ja3091615. ISSN 0002-7863.
7. Dey, Krishna Kanti; Das, Sambeeta; Poyton, Matthew F.; Sengupta, Samudra; Butler, Peter J.; Cremer, Paul S.; Sen, Ayusman (2014-12-23). "Chemotactic Separation of Enzymes". ACS Nano 8 (12): 11941–11949. doi:10.1021/nn504418u. ISSN 1936-0851.
8. Paxton, Walter F.; Kistler, Kevin C.; Olmeda, Christine C.; Sen, Ayusman; St. Angelo, Sarah K.; Cao, Yanyan; Mallouk, Thomas E.; Lammert, Paul E.; Crespi, Vincent H. (2004-10-01). "Catalytic Nanomotors:  Autonomous Movement of Striped Nanorods". Journal of the American Chemical Society 126 (41): 13424–13431. doi:10.1021/ja047697z. ISSN 0002-7863.
9. Liu, Ran; Sen, Ayusman (2011-12-21). "Autonomous Nanomotor Based on Copper–Platinum Segmented Nanobattery". Journal of the American Chemical Society 133 (50): 20064–20067. doi:10.1021/ja2082735. ISSN 0002-7863.
10. Pavlick, Ryan A.; Sengupta, Samudra; McFadden, Timothy; Zhang, Hua; Sen, Ayusman (2011-09-26). "A Polymerization-Powered Motor". Angewandte Chemie International Edition 50 (40): 9374–9377. doi:10.1002/anie.201103565. ISSN 1521-3773.
11. Vicsek, T.; Czirok, A.; Ben-Jacob, E.; Cohen, I.; Shochet, O. (1995). "Novel type of phase transition in a system of self-driven particles". Physical Review Letters 75 (6): 1226–1229. arXiv:cond-mat/0611743. Bibcode:1995PhRvL..75.1226V. doi:10.1103/PhysRevLett.75.1226. PMID 10060237.
12. Reynolds, C.W. (1987). "Flocks, herds and schools: A distributed behavioral model". Computer Graphics 21 (4): 25–34. doi:10.1145/37401.37406. ISBN 0897912276. CiteSeerX: 10.1.1.103.7187.
13. Czirók, A.; Vicsek, T. (2006). "Collective behavior of interacting self-propelled particles". Physica A 281: 17–29. arXiv:cond-mat/0611742. Bibcode:2000PhyA..281...17C. doi:10.1016/S0378-4371(00)00013-3.
14. Jadbabaie, A.; Lin, J.; Morse, A.S. (2003). "Coordination of groups of mobile autonomous agents using nearest neighbor rules". IEEE Transactions on Automatic Control 48 (6): 988–1001. doi:10.1109/TAC.2003.812781. CiteSeerX: 10.1.1.128.5326 – convergence proofs for the SPP model.
15. "Self driven particle model". Interactive simulations. University of Colorado. 2005. Retrieved 10 April 2011.
16. Uvarov, B. P. (1977). Behaviour, ecology, biogeography, population dynamics. Grasshopper and locust: a handbook of general acridology II. Cambridge University Press.
17. Symmons, P.M.; Cressman, K. (2001). "Desert locust guidelines: Biology and behaviour" (PDF). Rome: FAO.
18. Huepe, A.; Aldana, M. (2004). "Intermittency and clustering in a system of self-driven particles" (PDF). Physical Review Letters 92 (16): 168701 [4 pages]. Bibcode:2004PhRvL..92p8701H. doi:10.1103/PhysRevLett.92.168701.
19. Bhattacharya, K.; Vicsek, T. (2010). "Collective decision making in cohesive flocks". arXiv:1007.4453. Bibcode:2010NJPh...12i3019B. doi:10.1088/1367-2630/12/9/093019.
20. "Self-Propelled Particle System Improves Understanding Of Behavioral Patterns" (Press release). Medical News Today. 18 Sep 2010.
21. Somfai, E.; Czirok, A.; Vicsek, T. (1994). "Power-law distribution of landslides in an experiment on the erosion of a granular pile". Journal of Physics A: Mathematical and General 27 (20): L757–L763. Bibcode:1994JPhA...27L.757S. doi:10.1088/0305-4470/27/20/001.
22. "Bird flock decision-making revealed". Himalayan Times. 2010-09-14.
23. Gautrais, J.; Jost, C.; Theraulaz, G. (2008). "Key behavioural factors in a self-organised fish school model" (PDF). 45: 415–428.
24. Sugawara, K.; Sano, M.; Watanabe, T. (2009). "Nature of the order-disorder transition in the Vicsek model for the collective motion of self-propelled particles". Physical Review E 80 (5): 050103 [1–4]. Bibcode:2009PhRvE..80e0103B. doi:10.1103/PhysRevE.80.050103.
25. Chowdhury, D. (2006). "Collective effects in intra-cellular molecular motor transport: coordination, cooperation and competition". Physica A 372 (1): 84–95. arXiv:physics/0605053. Bibcode:2006PhyA..372...84C. doi:10.1016/j.physa.2006.05.005.
26. Helbing, D.; Farkas, I.; Vicsek, T. (2000). "Simulating dynamical features of escape panic". Nature 407 (6803): 487–490. arXiv:cond-mat/0009448. Bibcode:2000Natur.407..487H. doi:10.1038/35035023. PMID 11028994.
27. Helbing, D.; Keltsch, J.; Molnar, P. (1997). "Modelling the evolution of human trail systems". Nature 388 (6637): 47–50. arXiv:cond-mat/9805158. Bibcode:1997Natur.388...47H. doi:10.1038/40353. PMID 9214501.
28. Bickel, Thomas; Majee, Arghya; Würger, Alois (2013). "Flow pattern in the vicinity of self-propelling hot Janus particles". Physical Review E 88 (1). doi:10.1103/PhysRevE.88.012301. ISSN 1539-3755.

Further references

• Bertin, E.; Droz, M.; Grégoire, G. (2009). "Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis". Journal of Physics A 42 (44): 445001. arXiv:0907.4688. Bibcode:2009JPhA...42R5001B. doi:10.1088/1751-8113/42/44/445001.
• Czirók, A.; Stanley, H. E.; Vicsek, T. (1997). "Spontaneously ordered motion of self-propelled particles". Journal of Physics A 30 (5): 1375–1385. arXiv:cond-mat/0611741. Bibcode:1997JPhA...30.1375C. doi:10.1088/0305-4470/30/5/009.
• Czirók, A.; Barabási, A. L.; Vicsek, T. (1999). "Collective motion of self-propelled particles: Kinetic phase transition in one dimension". Physical Review Letters 82 (1): 209–212. arXiv:cond-mat/9712154. Bibcode:1999PhRvL..82..209C. doi:10.1103/PhysRevLett.82.209.
• Czirók, A.; Vicsek, T. (2001). "Flocking: collective motion of self-propelled particles". In Vicsek, T. Fluctuations and scaling in biology. Oxford University Press. pp. 177–209. ISBN 978-0-19-850790-1.
• D'Orsogna, M. R.; Chuang, Y. L.; Bertozzi, A. L.; Chayes, L. S. (2006). "Self-propelled particles with soft-core interactions: patterns, stability, and collapse" (PDF). Physical Review Letters 96 (10): 104302. Bibcode:2006PhRvL..96j4302D. doi:10.1103/PhysRevLett.96.104302.
• Levine, H.; Rappel, W. J.; Cohen, I. (2001). "Self-organization in systems of self-propelled particles". Physical Review E 63: 017101. arXiv:cond-mat/0006477. Bibcode:2001PhRvE..63a7101L. doi:10.1103/PhysRevE.63.017101.
• Mehandia, V.; Nott, P.R. (2008). "The collective dynamics of self-propelled particles". Journal of Fluid Mechanics 595: 239–264. arXiv:0707.1436. Bibcode:2008JFM...595..239M. doi:10.1017/S0022112007009184.
• Helbing, D. (2001). "The wonderful world of active many-particle systems". Advances in Solid State Physics 41. pp. 357–368. doi:10.1007/3-540-44946-9_29.
• Simha, R. A.; Ramaswamy, S. (2006). "Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles". Physical Review Letters 89 (5): 058101. arXiv:cond-mat/0108301. Bibcode:2002PhRvL..89e8101A. doi:10.1103/PhysRevLett.89.058101.
• Sumpter, D. J. T. (2010). "Chapter 5: Moving together". Collective Animal Behavior. Princeton University Press. ISBN 978-0-691-12963-1.
• Vicsek, T. (2010). "Statistical physics: Closing in on evaders". Nature 466 (7302): 43–44. Bibcode:2010Natur.466...43V. doi:10.1038/466043a.
• Yates, Christian A. (2007). On the dynamics and evolution of self-propelled particle models (PDF) (MSc thesis). Somerville College, University of Oxford.
• Yates, Christian A.; Baker, Ruth E.; Erban, Radek; Maini, Philip K. (Fall 2010). "Refining self-propelled particle models for collective behaviour" (PDF). Canadian Applied Mathematics Quarterly (Applied Mathematics Institute, University of Alberta) 18 (3).

External links

• Swarming desert locusts – Video clip from Planet Earth