Kinetic models coupled with Gaussian thermostats: macroscopic frameworks

Abstract

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.

Multiscale modeling of fibrosis – What's next? Reply to Comments on “Towards a unified approach in the modeling of fibrosis: A review with research perspective” by Martine Ben Amar and Carlo Bianca

“…This is a new topic that recently has gained much attention among scholars working in the field of the applied sciences. In particular the theoretical framework of the active particles proposed in [32] can be further generalized by considering a fractional-order derivative in the time and activity variables. The corresponding thermostatted framework with fractional order is thus a new paradigm for the derivation of specific models for fibrosis.…”

Towards a unified approach in the modeling of fibrosis: A review with research perspectives

“…Classically different kinds of scalings lead to different kinds of equations: parabolic and hyperbolic [172][173][174][175][176][177][178][179][180][181]. Specifically by defining a rescaling of time and space variables (low-field or high-field scaling), and under suitable technical assumptions, macroscopic equations typically of the mechanics continuum approach are derived.…”

On the Boltzmann gas mixture equation: Linking the kinetic and fluid regimes

“…In particular in [4,5,3] the authors have shown that the compressible Navier-Stokes equations is the governing hydrodynamic equations if the first order approximation is included when the Mach number is a constant and the Knudsen number goes to zero, and the hydrodynamic equation is the incompressible Navier-Stokes equations when the Mach number goes to zero with the same rate as the Knudsen number. A different approach has been recently proposed for the asymptotic limit of generalized Boltzmann-type equations in which the action of a thermostat has been taken into account, see [7], [8].…”