Kinetic models coupled with Gaussian thermostats: macroscopic frameworks

This paper deals with the modelling of complex systems composed of a large number of elements grouped into different functional subsystems. The modelling framework is that of the thermostatted kinetic theory, which consists of a set of nonlinear integro-differential equations. Another source of nonlinearity is the presence of a mathematical thermostat that ensures the control of the global energy of the system. Specifically, this paper is devoted to the derivation of evolution equations for the macroscopic variables (density and momentum) from the underlying description at the microscopic scale delivered by the thermostatted kinetic models. With this as the aim, hyperbolic-type and parabolic-type scalings of the thermostatted kinetics for the active particles model are performed and the resulting macroscopic equations are obtained. Finally, asymptotic methods are applied to the relaxation model.