Stationary states for the noncutoff Kac equation with a Gaussian thermostat

Bagland, Véronique; Wennberg, Bernt; Wondmagegne, Yosief

doi:10.1088/0951-7715/20/3/003

Cited by 19 publications

(10 citation statements)

References 9 publications

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“…The Boltzmann equation can be derived as the limit of infinitely many particles that obey the classical laws of mechanics. The Boltzmann equation for a system of particles that are subject to a thermostatted force field has been studied by Wondmagegne [53,52,51,5]. A formal derivation of a Boltzmann equation for a thermostatted system is carried out, and rigorous results on non-equilibrium stationary states are obtained for Kac's toy model of the Boltzmann equation [39].…”

Section: Introductionmentioning

confidence: 99%

Mass and energy balance laws derived from High-Field limits of thermostatted Boltzmann equations

Degond¹,

Wennberg²

2007

Communications in Mathematical Sciences

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Abstract. We derive coupled mass and energy balance laws from a High-Field limit of thermostatted Boltzmann equations. The starting point is a Boltzmann equation for elastic collisions subjected to a large force field. By adding a thermostat correction, it is possible to expand the solutions about a High-Field equilibrium obtained when balancing the thermostatted field drift operator with the elastic collision operator. To this aim, a hydrodynamic type scaling of the thermostatted Boltzmann equation is used, considering that the leading 'collision operator' actually consists of the combination of the thermostatted field operator and of the elastic collision operator. At leading order in the Knudsen number, the resulting model consist of coupled nonlinear first order partial differential equations. We investigate two cases. The first one is based on a one-dimensional BGK-type operator. The second one is three dimensional and concerns a Fokker-Planck collision operator. In both cases, we show that the resulting models are hyperbolic, thereby indicating that they might be appropriate for a use in physically realistic situations.

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Section: Introductionmentioning

confidence: 99%

Mass and energy balance laws derived from High-Field limits of thermostatted Boltzmann equations

Degond¹,

Wennberg²

2007

Communications in Mathematical Sciences

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“…Bearing the expression (19) in mind, u(t,k)/|u| 2+p is well-defined for all t>0, for 0 ≤ p<1, since the Fourier coefficients of u(t,k) up to order two vanish.…”

Section: Equation 26 Writesmentioning

confidence: 99%

“…It is worth stressing that to the best of our knowledge, this is the first time that this proof is gained for the thermostatted kinetic theory proposed in Bianca. 8 Similar problems have been investigated for the Kac equation [16][17][18][19][20] and the Boltzmann equation. [21][22][23][24] The main result strongly supports the applicability of the thermostatted kinetic theory for the modeling of nonequilibrium complex systems.…”

Section: Introductionmentioning

confidence: 99%

On the convergence toward nonequilibrium stationary states in thermostatted kinetic models

Bianca

Menale

2019

Math Methods in App Sciences

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A differential equation-based framework is suitable for the modeling of nonequilibrium complex systems if its solution is able to reach, as time goes to infinity, the related nonequilibrium steady states. The thermostatted kinetic theory framework has been recently proposed for the modeling of complex systems subjected to an external force field. The present paper is devoted to the mathematical proof of the convergence of the solutions of the thermostatted kinetic framework towards the related nonequilibrium stationary states. The proof of the main result is gained by employing the Fourier transform and distribution theory arguments.

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“…Recently the introduction of Gaussian thermostats in kinetic equations has gained much attention (Wennberg and Wondmagegne, 2004;Bagland et al, 2007;Degond and Wennberg, 2007). The Gaussian thermostat is a mathematical tool that allows to control the magnitude of energy of a system subjected to external force fields, see, among others, papers (Evans and Morriss, 1990;Morris and Dettmann, 1998;Jepps and Rondoni, 2006;Jepps et al, 2008;Jepps and Rondoni, 2010;Bianca, 2010;2011) for the recent investigations and new developments.…”

Section: Introductionmentioning

confidence: 99%

New Research Perspectives on Thermostatted Kinetic Models

Bianca¹

2015

Journal of Mathematics and Statistics

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This paper presents the new research directions that can be followed in the modeling of complex living systems by means of the tools of the thermostatted kinetic theory. The main aim is to invite the researchers that work in the field of applied mathematics to contribute to further developments of the theory with particular attention to applications in biology, crowds and swarm dynamics, economic and social systems.

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Stationary states for the noncutoff Kac equation with a Gaussian thermostat

Abstract: We study the stationary states of a Kac equation with a Gaussian thermostat in the case of a non-cutoff cross section. We investigate the existence, smoothness and uniqueness of the stationary states. The theoretical results are illustrated by some numerical simulations.

Cited by 19 publications

References 9 publications

Mass and energy balance laws derived from High-Field limits of thermostatted Boltzmann equations

Mass and energy balance laws derived from High-Field limits of thermostatted Boltzmann equations

On the convergence toward nonequilibrium stationary states in thermostatted kinetic models

New Research Perspectives on Thermostatted Kinetic Models

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