KRM
A Boltzmann-type model for market economy and its continuous trading limit
Marzia Bisi Giampiero Spiga
Kinetic & Related Models 2010, 3(2): 223-239 doi: 10.3934/krm.2010.3.223
In the frame of a kinetic Boltzmann-type approach to modeling market economies, a random conservative-in-the-mean scheme is proposed for binary transactions among agents. The scheme extends a very successful model recently introduced by Cordier, Pareschi and Toscani. Effects of the risky market on the overall output after the trade of each agent are accounted for by random variables affecting not only the wealth of that agent before the trade, but also the one of his partner. Variations induced by this generalization on steady distribution, existence of moments, and Pareto index are discussed. In particular, the continuous trading limit and the relevant limiting Fokker-Planck equation are commented on in detail.
keywords: Boltzmann equation; Wealth distribution; Econophysics.
KRM
On a kinetic BGK model for slow chemical reactions
Marzia Bisi Giampiero Spiga
Kinetic & Related Models 2011, 4(1): 153-167 doi: 10.3934/krm.2011.4.153
A recently proposed consistent BGK-type approach for chemically reacting gas mixtures is discussed, which accounts for the correct rates of transfer for mass, momentum and energy, and recovers the exact conservation equations and collision equilibria, including mass action law. In particular, the hydrodynamic limit is derived by a Chapman-Enskog procedure, and compared to existing results for the reactive and non-reactive cases.
keywords: Kinetic theory; Chemical reaction; BGK model.
KRM
A general consistent BGK model for gas mixtures
Alexander V. Bobylev Marzia Bisi Maria Groppi Giampiero Spiga Irina F. Potapenko
Kinetic & Related Models 2018, 11(6): 1377-1393 doi: 10.3934/krm.2018054

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 H-theorem. 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.

keywords: Kinetic theory Boltzmann equation BGK models, gas mixtures gas mixtures exchange rates
KRM
Kinetic approach to deflagration processes in a recombination reaction
Fiammetta Conforto Maria Groppi Roberto Monaco Giampiero Spiga
Kinetic & Related Models 2011, 4(1): 259-276 doi: 10.3934/krm.2011.4.259
Steady one-dimensional flame structure is investigated in a binary gas mixture made up by diatomic molecules and atoms, which undergo an irreversible exothermic two--steps reaction, a recombination process followed by inelastic scattering (de-excitation). A kinetic model at the Boltzmann level, accounting for chemical encounters as well as for mechanical collisions, is proposed and its main features are analyzed. In the case of collision dominated regime with slow recombination and fast de-excitation, the model is the starting point for a consistent derivation, via suitable asymptotic expansion of Chapman-Enskog type, of reactive fluid-dynamic Navier-Stokes equations. The resulting set of ordinary differential equations for the smooth steady deflagration profile is investigated in the frame of the qualitative theory of dynamical systems, and numerical results for the flame eigenvalue and for the main macroscopic observables are presented and briefly commented on for illustrative purposes.
keywords: Kinetic theory; Irreversible chemical reactions; Deflagration waves.
KRM
Flame structure from a kinetic model for chemical reactions
Marzia Bisi Maria Groppi Giampiero Spiga
Kinetic & Related Models 2010, 3(1): 17-34 doi: 10.3934/krm.2010.3.17
Steady one-dimensional flame structure is investigated in a binary mixture made up by two components of the same chemical species undergoing binary irreversible exothermic reactive encounters. A kinetic model at the Boltzmann level, accounting for chemical transitions as well as for mechanical collisions, is proposed and its main features are analyzed. In the case of slow chemical reactions and collision dominated regime, the model is the starting point for a consistent derivation, via suitable asymptotic expansion of Chapman-Enskog type, of reactive Navier-Stokes equations at the fluid-dynamic scale. The resulting set of ordinary differential equations is investigated in the frame of the qualitative theory of dynamical systems, and numerical results are presented and briefly commented on for illustrative purposes.
keywords: Kinetic theory; Irreversible chemical reactions; Deflagration waves.
KRM
Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics
Marzia Bisi Tommaso Ruggeri Giampiero Spiga
Kinetic & Related Models 2018, 11(1): 71-95 doi: 10.3934/krm.2018004

The aim of this paper is to compare different kinetic approaches to a polyatomic rarefied gas: the kinetic approach via a continuous energy parameter $I$ and the mixture-like one, based on discrete internal energy. We prove that if we consider only $6$ moments for a non-polytropic gas the two approaches give the same symmetric hyperbolic differential system previously obtained by the phenomenological Extended Thermodynamics. Both meaning and role of dynamical pressure become more clear in the present analysis.

keywords: Extended thermodynamics kinetic theory moment equations polyatomic gas dynamical pressure
DCDS-B
Qualitative analysis of kinetic-based models for tumor-immune system interaction
Martina Conte Maria Groppi Giampiero Spiga
Discrete & Continuous Dynamical Systems - B 2018, 23(9): 3663-3684 doi: 10.3934/dcdsb.2018060

A mathematical model, based on a mesoscopic approach, describing the competition between tumor cells and immune system in terms of kinetic integro-differential equations is presented. Four interacting components are considered, representing, respectively, tumors cells, cells of the host environment, cells of the immune system, and interleukins, which are capable to modify the tumor-immune system interaction and to contribute to destroy tumor cells. The internal state variable (activity) measures the capability of a cell of prevailing in a binary interaction. Under suitable assumptions, a closed set of autonomous ordinary differential equations is then derived by a moment procedure and two three-dimensional reduced systems are obtained in some partial quasi-steady state approximations. Their qualitative analysis is finally performed, with particular attention to equilibria and their stability, bifurcations, and their meaning. Results are obtained on asymptotically autonomous dynamical systems, and also on the occurrence of a particular backward bifurcation.

keywords: Kinetic models Kinetic models macroscopic closures asymptotically autonomous systems stability forward and backward bifurcations

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