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Entropy 2016, 18(1), 27; doi:10.3390/e18010027

Interacting Brownian Swarms: Some Analytical Results

STI/IMT/LPM, Ecole Polytechnique Fédérale de Lausanne, Station 17 (Batiment BM), CH-1015 Lausanne, Switzerland
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Academic Editor: Giorgio Sonnino
Received: 18 November 2015 / Revised: 7 January 2016 / Accepted: 11 January 2016 / Published: 14 January 2016
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Abstract

We consider the dynamics of swarms of scalar Brownian agents subject to local imitation mechanisms implemented using mutual rank-based interactions. For appropriate values of the underlying control parameters, the swarm propagates tightly and the distances separating successive agents are iid exponential random variables. Implicitly, the implementation of rank-based mutual interactions, requires that agents have infinite interaction ranges. Using the probabilistic size of the swarm’s support, we analytically estimate the critical interaction range below that flocked swarms cannot survive. In the second part of the paper, we consider the interactions between two flocked swarms of Brownian agents with finite interaction ranges. Both swarms travel with different barycentric velocities, and agents from both swarms indifferently interact with each other. For appropriate initial configurations, both swarms eventually collide (i.e., all agents interact). Depending on the values of the control parameters, one of the following patterns emerges after collision: (i) Both swarms remain essentially flocked, or (ii) the swarms become ultimately quasi-free and recover their nominal barycentric speeds. We derive a set of analytical flocking conditions based on the generalized rank-based Brownian motion. An extensive set of numerical simulations corroborates our analytical findings. View Full-Text
Keywords: rank-based Brownian motions; heterogeneous swarm; hybrid atlas model; interactions between swarms; leader–follower dynamics; analytical flocking conditions; numerical simulations rank-based Brownian motions; heterogeneous swarm; hybrid atlas model; interactions between swarms; leader–follower dynamics; analytical flocking conditions; numerical simulations
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Sartoretti, G.; Hongler, M.-O. Interacting Brownian Swarms: Some Analytical Results. Entropy 2016, 18, 27.

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