Microscopic, mesoscopic, and macroscopic descriptions of the Euler alignment system with adaptive communication strength

The $ {\mathrm{s}} $-model, introduced by the first author in [Environmental averaging. EMS Surv. Math. Sci., 11 (2024), no. 2, 277-413], generalizes the Cucker-Smale model by allowing a broader class of velocity averaging protocols, while preserving the 1D conservation law: $ \partial_t e + \partial_x (ue) = 0 $, $ e = \partial_x u + {\mathrm{s}} $. As a continuation of our previous joint work, [Well-posedness and long time behavior of the Euler Alignment System with adaptive communication strength, accepted at the Abel Symposium Proceedings, also arXiv:2310.00269, 2023], this paper aims to establish the physical foundations of the $ {\mathrm{s}} $-model by deriving and justifying its microscopic and mesoscopic formulations. A distinctive feature of the microscopic system is that it is a discrete-continuous system: the position and velocity of the particles are discrete, while $ {\mathrm{s}} $ is an active continuum scalar field. We rigorously derive the mesoscopic description via the mean-field limit and further obtain the macroscopic equations in the monokinetic and Maxwellian regimes. Additionally, we analyze the long-time behavior of the kinetic Fokker-Planck-Alignment equation by establishing the relaxation to the Maxwellian in 1D for Favre-averaged velocity fields. As a supplement to our previous numerical results, we present particle simulations which reinforce the qualitative similarities between the $ {\mathrm{s}} $-model and the Motsch-Tadmor model.