| Popis: |
The $\mathrm{s}$-model, introduced in arXiv:2211.00117, is an alignment model with the property that the strength of the alignment force, $\mathrm{s}$, is transported along an averaged velocity field. Inspired by the 1D threshold regularity criterion for the Cucker-Smale model in terms of the so-called $e$-quantity, $e = \partial_x u + \rho \ast \phi$, the transport of the strength in the $\mathrm{s}$-model was designed so that it possesses its own $e$-quantity, $e = \partial_x u + \mathrm{s}$. The $e$-quantity lends the $\mathrm{s}$-model to a similar threshold regularity criterion and long-time behavior in 1D, arXiv:2310.00269. The goal of this paper is to put the $\mathrm{s}$-model on more firm physical grounds by formulating and justifying the microscopic and mesoscopic descriptions from which it arises. A distinctive feature of the microscopic system is that it is a discrete-continuous system: the position and velocity of the particles are discrete objects, while the strength is an active continuum scalar function. We establish a rigorous passage from the microscopic to the mesoscopic description via the Mean Field Limit and a passage from the mesoscopic to the macroscopic description in the monokinetic and Maxwellian limiting regimes. We present a survey of such results for the Cucker-Smale model and explain how to extend these arguments to the $\mathrm{s}$-model, where the strength of the alignment force is transported. We also address the long-time behavior of the kinetic Fokker-Planck-Alignment equation by establishing the relaxation to the Maxwellian in 1D when the velocity averaging is the Favre averaging. As a supplement to the numerical results already presented in arXiv:2310.00269, we provide additional numerical evidence, via a particle simulation, that the $\mathrm{s}$-model behaves qualitatively like Motsch-Tadmor. |
