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ABSTRACT There are many instances of collective behaviors in the natural world. For example, eukaryotic cells coordinate their motion to heal wounds; bacteria swarm during colony expansion; defects in alignment in growing bacterial populations lead to biofilm growth; and birds move within dynamic flocks. Although the details of how these groups behave vary across animals and species, they share the same qualitative feature: they exhibit collective behaviors that are not simple extensions of details associated with the motion of an individual. To learn more about these biological systems, we propose studying these systems through the lens of the foundational Vicsek model. Here, we present the process of building this computational model from scratch in a tutorial format that focuses on building the appropriate skills of an undergraduate student. In doing so, an undergraduate student should be able to work alongside this article, the corresponding tutorial, and the original manuscript of the Vicsek model to build their own model. We conclude by summarizing some of the current work involving computational modeling of flocking with Vicsek-type models.
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The "hot spots" conjecture on the Vicsek set
Abstract We prove the “hot spots” conjecture on the Vicsek set. Specifically, we will show that every eigenfunction of the second smallest eigenvalue of the Neumann Laplacian on the Vicsek set attains its maximum and minimum on the boundary.
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- Award ID(s):
- 1743819
- PAR ID:
- 10088777
- Date Published:
- Journal Name:
- Demonstratio Mathematica
- Volume:
- 52
- Issue:
- 1
- ISSN:
- 2391-4661
- Page Range / eLocation ID:
- 61 to 81
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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In this work, we explore how the emergence of collective motion in a system of particles is influenced by the structure of their domain. Using the Vicsek model to generate flocking, we simulate two-dimensional systems that are confined based on varying obstacle arrangements. The presence of obstacles alters the topological structure of the domain where collective motion occurs, which, in turn, alters the scaling behavior. We evaluate these trends by considering the scaling exponent and critical noise threshold for the Vicsek model, as well as the associated diffusion properties of the system. We show that obstacles tend to inhibit collective motion by forcing particles to traverse the system based on curved trajectories that reflect the domain topology. Our results highlight key challenges related to the development of a more comprehensive understanding of geometric structure's influence on collective behavior.more » « less
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null (Ed.)The emergence of orientational order plays a central role in active matter theory and is deeply based in the study of active systems with a velocity alignment mechanism, whose most prominent example is the so-called Vicsek model. Such active systems have been used to describe bird flocks, bacterial swarms, and active colloidal systems, among many other examples. Under the assumption that the large-scale properties of these models remain unchanged as long as the polar symmetry of the interactions is not affected, implementations have been performed using, out of convenience, either additive or non-additive interactions; the latter are found for instance in the original formulation of the Vicsek model. Here, we perform a careful analysis of active systems with velocity alignment, comparing additive and non-additive interactions, and show that the macroscopic properties of these active systems are fundamentally different. Our results call into question our current understanding of the onset of order in active systems.more » « less
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In this paper we prove that several natural approaches to Sobolev spaces coincide on the Vicsek fractal. More precisely, we show that the metric approach of Korevaar-Schoen, the approach by limit of discrete \(p\)-energies and the approach by limit of Sobolev spaces on cable systems all yield the same functional space with equivalent norms for \(p>1\). As a consequence we prove that the Sobolev spaces form a real interpolation scale. We also obtain \(L^p\)-Poincaré inequalities for all values of \(p \ge 1\).more » « less
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Active matter is differentiated from conventional passive matter due to its unique capability of locally consuming fuels to generate kinetic energy. Such a unique feature of active matter has led to unprecedented phenomena and associated applications. While active matter has been developed for decades, its significance is not recognized by the public. To remedy this gap, we developed an online teaching module introducing collective dynamics of active matter, targeting high school and undergraduate students. The collective dynamics were illustrated via the Vicsek model-based simulation because it reveals the collective dynamics of active matter with one simple rule: nearest-neighbor alignment. With this rule, the simulation demonstrated the collective motion of active matter particles depended on particle number, radius of neighbor aligning, and noise that disturbed alignment. To allow students to hands-on experience the simulation, we developed a graphical user interface, allowing users to perform the Vicsek simulation without a programming background. The simulation and teaching module are available on an online platform: The Partnership for Integration of Computation into Undergraduate Physics, allowing teachers in the US to bring the active matter lecture to their classrooms.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1515/dema-2019-0003
