A recently proposed consistent approach for elastically scattering gas mixtures, of the type introduced by Bhatnagar, Gross, and Krook (BGK), has been extended to deal with a four species gas undergoing reversible bimolecular chemical reactions. The single BGK collision operator introduced for each species must take into account also transfer of mass and of energy of chemical bond. Suitable auxiliary fields have then to be introduced not only for temperatures and velocities, but also for densities, in order to fulfill correctly balance equations for mass, momentum, and total energy. The exact collision equilibrium, satisfying the mass action law of chemistry, is also recovered, and the proper choice of collision frequencies is discussed. Preliminary numerical results for the relaxation problem in space-homogeneous conditions are reported and briefly commented on.
We introduce and discuss kinetic models for wealth distribution in a simple market economy, which are able to reproduce the salient features of the wealth distribution by including taxes to each trading process and redistributing the collected money among the population according to a given criterion. Our analysis gives a theoretical basis to some recent research that analyzed discrete simplified models for the exploitation of finite resources by interacting agents, where each agent receives a random fraction of the available resources. It is shown that in general the redistribution is able to modify the Pareto index, and that this modification can be quantified in terms of the redistribution operator.
An ellipsoidal BGK model is proposed for a binary mixture of rarefied gases in the frame of kinetic theory. It fulfils the crucial properties of the actual Boltzmann equation (collision invariants, equilibria, entropy dissipation), and introduces a further constraint on velocity equalization of the two species. The model features two disposable relaxation parameters which can be used to fit exactly, in the continuum limit, Fick's law for diffusion velocities and Newton's law for the viscous stress in the relevant set of Navier-Stokes equations. Positivity of temperature fields is guaranteed by a physically meaningful restriction on the parameters themselves.
We propose a kinetic model of BGK type for a gas mixture of an arbitrary number of species with arbitrary collision law. The model features the same structure of the corresponding Boltzmann equations and fulfils all consistency requirements concerning conservation laws, equilibria, and Htheorem. Comparison is made to existing BGK models for mixtures, and the achieved improvements are commented on. Finally, possible application to the case of Coulomb interaction is briefly discussed.
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