Abstract
The paper treats an agent-based model with averaging dynamics to which we refer as the K-averaging model. Broadly speaking, our model can be added to the growing list of dynamics exhibiting self-organization such as the well-known Vicsek-type models (Aldana et al. in: Phys Rev Lett 98(9):095702, 2007; Aldana and Huepe in: J Stat Phys 112(1–2):135–153, 2003; Pimentel in: Phys. Rev. E 77(6):061138, 2008). In the K-averaging model, each of the N particles updates their position by averaging over K randomly selected particles with additional noise. To make the K-averaging dynamics more tractable, we first establish a propagation of chaos type result in the limit of infinite particle number (i.e. \(N \rightarrow \infty \)) using a martingale technique. Then, we prove the convergence of the limit equation toward a suitable Gaussian distribution in the sense of Wasserstein distance as well as relative entropy. We provide additional numerical simulations to illustrate both results.
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Acknowledgements
It is a pleasure to thank my Ph.D. advisor Sébastien Motsch for his tremendous help on various portions of this manuscript, and I thank the anonymous reviewer for providing helpful comments on earlier drafts of the manuscript.
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Communicated by Irene Giardina.
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Cao, F. K-Averaging Agent-Based Model: Propagation of Chaos and Convergence to Equilibrium. J Stat Phys 184, 18 (2021). https://doi.org/10.1007/s10955-021-02807-0
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DOI: https://doi.org/10.1007/s10955-021-02807-0