From discrete Cucker-Smale model to continuous Cucker-Smale model in a temperature field

We study emergent dynamics of the discrete and continuous Cucker-Smale (C-S) models in a self-consistent temperature field and provide refined monocluster flocking estimates. More precisely, we present a sufficient framework leading to the monocluster flocking toward the constant common temperature and velocity exponentially fast. This improves the authors’ earlier work where existence of common temperature and velocity are not identified. Our proposed sufficient framework is formulated in terms of the initial data and system parameters. Using the refined flocking, we also provide a uniform-in-time convergence from the discrete model to the continuous model under some suitable compatibility condition on the asymptotic velocities for the discrete and continuous model, as the time step tends to zero. We also provide several numerical examples and compare them with analytical results.