Propagation of chaos for the Cucker-Smale systems under heavy tail communication

• Published in: Communications in partial differential equations Vol. 47; no. 9; pp. 1883 - 1906
• Main Authors: Nguyen, Vinh, Shvydkoy, Roman
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 02.09.2022; Taylor & Francis Ltd
• Subjects: Alignment; Collective behavior; Communications systems; Cucker-Smale; Kernels; Liouville equations; mean-field limit; Propagation; propagation of chaos; Time dependence
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2022.2091454

In this work, we study propagation of chaos for solutions of the Liouville equation derived from 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 to the product of k solutions of the corresponding Vlasov-Alignment equation, Specifically, the following estimate holds in terms of Wasserstein-2 metric (1) 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. We recast the method in the kinetic setting and show that the bound ( 1 ) 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, although those bounds hold for general kernels. The main message of our work is that flocking dynamics improves the rate considerably.