1
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Lizárraga JUF, O'Keeffe KP, de Aguiar MAM. Order, chaos, and dimensionality transition in a system of swarmalators. Phys Rev E 2024; 109:044209. [PMID: 38755840 DOI: 10.1103/physreve.109.044209] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2023] [Accepted: 03/20/2024] [Indexed: 05/18/2024]
Abstract
Similarly to sperm, where individuals self-organize in space while also striving for coherence in their tail swinging, several natural and engineered systems exhibit the emergence of swarming and synchronization. The arising and interplay of these phenomena have been captured by collectives of hypothetical particles named swarmalators, each possessing a position and a phase whose dynamics are affected reciprocally and also by the space-phase states of their neighbors. In this work, we introduce a solvable model of swarmalators able to move in two-dimensional spaces. We show that several static and active collective states can emerge and derive necessary conditions for each to show up as the model parameters are varied. These conditions elucidate, in some cases, the displaying of multistability among states. Notably, in the active regime, the system exhibits hyperchaos, maintaining spatial correlation under certain conditions and breaking it under others on what we interpret as a dimensionality transition.
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Affiliation(s)
- Joao U F Lizárraga
- Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, Unicamp 13083-970, Campinas, São Paulo, Brazil
| | - Kevin P O'Keeffe
- Senseable City Lab, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Marcus A M de Aguiar
- Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, Unicamp 13083-970, Campinas, São Paulo, Brazil
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2
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Blum N, Li A, O'Keeffe K, Kogan O. Swarmalators with delayed interactions. Phys Rev E 2024; 109:014205. [PMID: 38366397 DOI: 10.1103/physreve.109.014205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2022] [Accepted: 09/15/2023] [Indexed: 02/18/2024]
Abstract
We investigate the effects of delayed interactions in a population of "swarmalators," generalizations of phase oscillators that both synchronize in time and swarm through space. We discover two steady collective states: a state in which swarmalators are essentially motionless in a disk arranged in a pseudocrystalline order, and a boiling state in which the swarmalators again form a disk, but now the swarmalators near the boundary perform boiling-like convective motions. These states are reminiscent of the beating clusters seen in photoactivated colloids and the living crystals of starfish embryos.
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Affiliation(s)
- Nicholas Blum
- California Polytechnic State University, San Luis Obispo, California 93407, USA
| | - Andre Li
- Department of Physics, California State University, East Bay, California 94542, USA
| | - Kevin O'Keeffe
- Senseable City Lab, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Oleg Kogan
- California Polytechnic State University, San Luis Obispo, California 93407, USA
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3
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Rombouts J, Elliott J, Erzberger A. Forceful patterning: theoretical principles of mechanochemical pattern formation. EMBO Rep 2023; 24:e57739. [PMID: 37916772 DOI: 10.15252/embr.202357739] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2023] [Revised: 09/21/2023] [Accepted: 09/27/2023] [Indexed: 11/03/2023] Open
Abstract
Biological pattern formation is essential for generating and maintaining spatial structures from the scale of a single cell to tissues and even collections of organisms. Besides biochemical interactions, there is an important role for mechanical and geometrical features in the generation of patterns. We review the theoretical principles underlying different types of mechanochemical pattern formation across spatial scales and levels of biological organization.
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Affiliation(s)
- Jan Rombouts
- Cell Biology and Biophysics Unit, European Molecular Biology Laboratory (EMBL), Heidelberg, Germany
- Developmental Biology Unit, European Molecular Biology Laboratory (EMBL), Heidelberg, Germany
| | - Jenna Elliott
- Cell Biology and Biophysics Unit, European Molecular Biology Laboratory (EMBL), Heidelberg, Germany
- Department of Physics and Astronomy, Heidelberg University, Heidelberg, Germany
| | - Anna Erzberger
- Cell Biology and Biophysics Unit, European Molecular Biology Laboratory (EMBL), Heidelberg, Germany
- Department of Physics and Astronomy, Heidelberg University, Heidelberg, Germany
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4
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Hao B, Zhong M, O'Keeffe K. Attractive and repulsive interactions in the one-dimensional swarmalator model. Phys Rev E 2023; 108:064214. [PMID: 38243440 DOI: 10.1103/physreve.108.064214] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/06/2023] [Accepted: 11/30/2023] [Indexed: 01/21/2024]
Abstract
We study a population of swarmalators, mobile variants of phase oscillators, which run on a ring and have both attractive and repulsive interactions. This one-dimensional (1D) swarmalator model produces several of collective states: the standard sync and async states as well as a splaylike "polarized" state and several unsteady states such as active bands or swirling. The model's simplicity allows us to describe some of the states analytically. The model can be considered as a toy model for real-world swarmalators such as vinegar eels and sperm which swarm in quasi-1D geometries.
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Affiliation(s)
- Baoli Hao
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA
| | - Ming Zhong
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA
| | - Kevin O'Keeffe
- Senseable City Lab, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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5
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Sar GK, O'Keeffe K, Ghosh D. Swarmalators on a ring with uncorrelated pinning. CHAOS (WOODBURY, N.Y.) 2023; 33:111103. [PMID: 37938924 DOI: 10.1063/5.0177024] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2023] [Accepted: 10/10/2023] [Indexed: 11/10/2023]
Abstract
We present a case study of swarmalators (mobile oscillators) that move on a 1D ring and are subject to pinning. Previous work considered the special case where the pinning in space and the pinning in the phase dimension were correlated. Here, we study the general case where the space and phase pinning are uncorrelated, both being chosen uniformly at random. This induces several new effects, such as pinned async, mixed states, and a first-order phase transition. These phenomena may be found in real world swarmalators, such as systems of vinegar eels, Janus matchsticks, electrorotated Quincke rollers, or Japanese tree frogs.
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Affiliation(s)
- Gourab Kumar Sar
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Kevin O'Keeffe
- Senseable City Lab, Massachusetts Institute of Technology, Cambridge Massachusetts 02139, USA
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
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6
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Quillen AC. Robust formation of metachronal waves in directional chains of phase oscillators. Phys Rev E 2023; 107:034401. [PMID: 37073033 DOI: 10.1103/physreve.107.034401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2022] [Accepted: 02/16/2023] [Indexed: 04/20/2023]
Abstract
Biological systems can rely on collective formation of a metachronal wave in an ensemble of oscillators for locomotion and for fluid transport. We consider one-dimensional chains of phase oscillators with nearest-neighbor interactions, connected in a loop and with rotational symmetry, so each oscillator resembles every other oscillator in the chain. Numerical integrations of the discrete phase oscillator systems and a continuum approximation show that directional models (those that do not obey reversal symmetry), can exhibit instability to short wavelength perturbations but only in regions where the slope in phase has a particular sign. This causes short wavelength perturbations to develop that can vary the winding number that describes the sum of phase differences across the loop and the resulting metachronal wave speed. Numerical integrations of stochastic directional phase oscillator models show that even a weak level of noise can seed instabilities that resolve into metachronal wave states.
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Affiliation(s)
- A C Quillen
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
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7
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Ceron S, O’Keeffe K, Petersen K. Diverse behaviors in non-uniform chiral and non-chiral swarmalators. Nat Commun 2023; 14:940. [PMID: 36806287 PMCID: PMC9941214 DOI: 10.1038/s41467-023-36563-4] [Citation(s) in RCA: 7] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2022] [Accepted: 02/06/2023] [Indexed: 02/22/2023] Open
Abstract
We study the emergent behaviors of a population of swarming coupled oscillators, dubbed swarmalators. Previous work considered the simplest, idealized case: identical swarmalators with global coupling. Here we expand this work by adding more realistic features: local coupling, non-identical natural frequencies, and chirality. This more realistic model generates a variety of new behaviors including lattices of vortices, beating clusters, and interacting phase waves. Similar behaviors are found across natural and artificial micro-scale collective systems, including social slime mold, spermatozoa vortex arrays, and Quincke rollers. Our results indicate a wide range of future use cases, both to aid characterization and understanding of natural swarms, and to design complex interactions in collective systems from soft and active matter to micro-robotics.
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Affiliation(s)
- Steven Ceron
- grid.5386.8000000041936877XSibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853 USA ,grid.116068.80000 0001 2341 2786Computer Science and Artificial Intelligence Lab, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
| | - Kevin O’Keeffe
- grid.116068.80000 0001 2341 2786Senseable City Lab, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
| | - Kirstin Petersen
- Electrical and Computer Engineering, Cornell University, 136 Hoy Road, Ithaca, NY, 14853, USA.
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8
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Sar GK, Ghosh D, O'Keeffe K. Pinning in a system of swarmalators. Phys Rev E 2023; 107:024215. [PMID: 36932525 DOI: 10.1103/physreve.107.024215] [Citation(s) in RCA: 9] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/04/2022] [Accepted: 02/01/2023] [Indexed: 03/19/2023]
Abstract
We study a population of swarmalators (swarming/mobile oscillators) which run on a ring and are subject to random pinning. The pinning represents the tendency of particles to stick to defects in the underlying medium which competes with the tendency to sync and swarm. The result is rich collective behavior. A highlight is low dimensional chaos which in systems of ordinary, Kuramoto-type oscillators is uncommon. Some of the states (the phase wave and split phase wave) resemble those seen in systems of Janus matchsticks or Japanese tree frogs. The others (such as the sync and unsteady states) may be observable in systems of vinegar eels, electrorotated Quincke rollers, or other swarmalators moving in disordered environments.
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Affiliation(s)
- Gourab Kumar Sar
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Kevin O'Keeffe
- Senseable City Lab, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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9
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Quillen AC, Peshkov A, Chakrabarti B, Skerrett N, McGaffigan S, Zapiach R. Fluid circulation driven by collectively organized metachronal waves in swimming T. aceti nematodes. Phys Rev E 2022; 106:064401. [PMID: 36671190 DOI: 10.1103/physreve.106.064401] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2022] [Accepted: 11/17/2022] [Indexed: 06/17/2023]
Abstract
Recent experiments have shown that the nematode T. aceti can assemble into collectively undulating groups at the edge of fluid drops. This coordinated state consists of metachronal waves and drives fluid circulation inside the drop. We find that the circulation velocity is about 2 mm/s and nearly half the speed of the metachronal wave. We develop a quasi-two-dimensional hydrodynamics model using the Stokes flow approximation. The periodic motion of the nematodes constitute our moving boundary condition that drives the flow. Our model suggests that large-amplitude excursions of the nematode tails produce the fluid circulation. We discuss the constraints on containers that would enhance fluid motion, which could be used in the future design of on demand flow generating systems.
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Affiliation(s)
- A C Quillen
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
| | - A Peshkov
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
| | - Brato Chakrabarti
- Center for Computational Biology, Flatiron Institute, New York, New York 10010, USA
| | - Nathan Skerrett
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
| | - Sonia McGaffigan
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
| | - Rebeca Zapiach
- Department of Mechanical Engineering, University of Rochester, Rochester, New York 14627, USA
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10
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Yoon S, O'Keeffe KP, Mendes JFF, Goltsev AV. Sync and Swarm: Solvable Model of Nonidentical Swarmalators. PHYSICAL REVIEW LETTERS 2022; 129:208002. [PMID: 36462001 DOI: 10.1103/physrevlett.129.208002] [Citation(s) in RCA: 11] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/18/2022] [Accepted: 10/13/2022] [Indexed: 06/17/2023]
Abstract
We study a model of nonidentical swarmalators, generalizations of phase oscillators that both sync in time and swarm in space. The model produces four collective states: asynchrony, sync clusters, vortexlike phase waves, and a mixed state. These states occur in many real-world swarmalator systems such as biological microswimmers, chemical nanomotors, and groups of drones. A generalized Ott-Antonsen ansatz provides the first analytic description of these states and conditions for their existence. We show how this approach may be used in studies of active matter and related disciplines.
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Affiliation(s)
- S Yoon
- Departamento de Física da Universidade de Aveiro and I3N, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal
| | - K P O'Keeffe
- Senseable City Lab, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - J F F Mendes
- Departamento de Física da Universidade de Aveiro and I3N, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal
| | - A V Goltsev
- Departamento de Física da Universidade de Aveiro and I3N, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal
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11
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O'Keeffe K, Hong H. Swarmalators on a ring with distributed couplings. Phys Rev E 2022; 105:064208. [PMID: 35854595 DOI: 10.1103/physreve.105.064208] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2022] [Accepted: 05/25/2022] [Indexed: 06/15/2023]
Abstract
We study a simple model of identical "swarmalators," generalizations of phase oscillators that swarm through space. We confine the movements to a one-dimensional (1D) ring and consider distributed (nonidentical) couplings; the combination of these two effects captures an aspect of the more realistic two-dimensional swarmalator model. We discover several collective states which we describe analytically. These states imitate the behavior of vinegar eels, catalytic microswimmers, and other swarmalators which move on quasi-1D rings.
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Affiliation(s)
- Kevin O'Keeffe
- Senseable City Lab, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Hyunsuk Hong
- Department of Physics and Research Institute of Physics and Chemistry, Jeonbuk National University, Jeonju 54896, Korea
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
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12
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Zhou W, Peralta JD, Hao Z, Gravish N. Lateral contact yields longitudinal cohesion in active undulatory systems. Phys Rev E 2022; 105:054604. [PMID: 35706245 DOI: 10.1103/physreve.105.054604] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/10/2021] [Accepted: 04/11/2022] [Indexed: 06/15/2023]
Abstract
Many animals and robots move using undulatory motion of their bodies. When the bodies are in close proximity undulatory motion can lead to novel collective behavior such as gait synchronization, spatial reconfiguration, and clustering. Here we study the role of contact interactions between model undulatory swimmers: three-link robots in experiment and multilink swimmers in simulation. The undulatory gait of each swimmer is generated through a time-dependent sinusoidal-like waveform which has a fixed phase offset, ϕ. By varying the phase relationship between neighboring swimmers we seek to study how contact forces and planar configurations are governed by the phase difference between neighboring swimmers. We find that undulatory actuation in close proximity drives neighboring swimmers into planar equilibrium configurations that depend on the actuation phase difference. We propose a model for stable planar configurations of nearest-neighbor undulatory swimmers which we call the gait compatibility condition, which is the set of planar and phase configurations in which no collisions occur. Robotic experiments with two, three, and four swimmers exhibit good agreement with the compatibility model. To study the contact forces and the time-averaged equilibrium between undulatory systems we perform simulations. To probe the interaction potential between undulatory swimmers we apply a small force to each swimmer longitudinally to separate them from the compatible configuration and we measure their steady-state displacement. These studies reveal that undulatory swimmers in close proximity exhibit attractive longitudinal interaction forces that drive the swimmers from incompatible to compatible configurations. This system of undulatory swimmers provides new insight into active-matter systems which move through body undulation. In addition to the importance of velocity and orientation coherence in active-matter swarms, we demonstrate that undulatory phase coherence is also important for generating stable, cohesive group configurations.
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Affiliation(s)
- Wei Zhou
- Mechanical and Aerospace Engineering, University of California, San Diego, California 92093, USA
| | - Jaquelin Dezha Peralta
- Mechanical and Aerospace Engineering, University of California, San Diego, California 92093, USA
| | - Zhuonan Hao
- Mechanical and Aerospace Engineering, University of California, San Diego, California 92093, USA
| | - Nick Gravish
- Mechanical and Aerospace Engineering, University of California, San Diego, California 92093, USA
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13
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Hao Z, Zhou W, Gravish N. Proprioceptive feedback design for gait synchronization in collective undulatory robots. Adv Robot 2022. [DOI: 10.1080/01691864.2022.2050810] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/01/2022]
Affiliation(s)
- Zhuonan Hao
- Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA, USA
| | - Wei Zhou
- Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA, USA
| | - Nick Gravish
- Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA, USA
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14
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Peshkov A, McGaffigan S, Quillen AC. Synchronized oscillations in swarms of nematode Turbatrix aceti. SOFT MATTER 2022; 18:1174-1182. [PMID: 35029257 DOI: 10.1039/d1sm01572a] [Citation(s) in RCA: 13] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
There is a recent surge of interest in the behavior of active particles that can at the same time align their direction of movement and synchronize their oscillations, known as swarmalators. While theoretical and numerical models of such systems are now abundant, no real-life examples have been shown to date. We present an experimental investigation of the collective motion of the nematode Turbatrix aceti that self-propel by body undulation. We discover that these nematodes can synchronize their body oscillations, forming striking traveling metachronal waves, which produces strong fluid flows. We uncover that the location and strength of this collective state can be controlled through the shape of the confining structure; in our case the contact angle of a droplet. This opens a way for producing controlled work such as on-demand flows or displacement of objects. We illustrate this by showing that the force generated by this state is sufficient to change the physics of evaporation of fluid droplets, by counteracting the surface-tension force, which allow us to estimate its strength. The relatively large size and ease of culture make Turbatrix aceti a promising model organism for experimental investigation of swarming and oscillating active matter capable of producing controllable work.
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Affiliation(s)
- Anton Peshkov
- Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA.
| | - Sonia McGaffigan
- Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA.
| | - Alice C Quillen
- Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA.
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15
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O'Keeffe K, Ceron S, Petersen K. Collective behavior of swarmalators on a ring. Phys Rev E 2022; 105:014211. [PMID: 35193221 DOI: 10.1103/physreve.105.014211] [Citation(s) in RCA: 17] [Impact Index Per Article: 8.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/04/2021] [Accepted: 12/07/2021] [Indexed: 06/14/2023]
Abstract
We study the collective behavior of swarmalators, generalizations of phase oscillators that both sync and swarm, confined to move on a one-dimensional (1D) ring. This simple model captures the essence of movement in two or three dimensions, but has the benefit of being solvable: most of the collective states and their bifurcations can be specified exactly. The model also captures the behavior of real-world swarmalators which swarm in quasi-1D rings such as bordertaxic vinegar eels and sperm.
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Affiliation(s)
- Kevin O'Keeffe
- Senseable City Lab, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Steven Ceron
- Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853, USA
| | - Kirstin Petersen
- Department of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA
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