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- W4200635942 abstract "In this work we study propagation of chaos for solutions of the Liouville equation for the classical discrete Cucker-Smale system. Assuming that the communication kernel satisfies the heavy tail condition -- known to be necessary to induce exponential alignment -- we obtain a linear in time convergence rate of the $k$-th marginals $f^{(k)}$ to the product of $k$ solutions of the corresponding Vlasov-Alignment equation, $f^{otimes k}$. Specifically, the following estimate holds in terms of Wasserstein-2 metric begin{equation}label{e:abst} mathcal{W}_2(f^{(k)}_t,f^{otimes k}_t) leq C k^{1/2} minleft{ 1, frac{t}{sqrt{N}} right}. end{equation} For systems with the Rayleigh-type friction and self-propulsion force, we obtain a similar result for sectorial solutions. Such solutions are known to align exponentially fast via the method of Grassmannian reduction, cite{LRS-friction}. We recast the method in the kinetic settings and show that the bound eqref{e:abst} persists but with the quadratic dependence on time. In both the forceless and forced cases, the result represents an improvement over the exponential bounds established earlier in the work of Natalini and Paul, cite{NP2021-orig}, although those bounds hold for general kernels. The main message of our work is that flocking dynamics improves the rate considerably." @default.
- W4200635942 created "2021-12-31" @default.
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- W4200635942 date "2021-12-08" @default.
- W4200635942 modified "2023-09-23" @default.
- W4200635942 title "Propagation of Chaos for the Cucker-Smale Systems under heavy tail communication" @default.
- W4200635942 doi "https://doi.org/10.48550/arxiv.2112.04437" @default.
- W4200635942 hasPublicationYear "2021" @default.
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