Well-Posedness and Large Time Behaviour for the Non-cutoff Kac Equation with a Gaussian Thermostat

Bagland, Véronique

doi:10.1007/s10955-009-9872-4

Cited by 14 publications

(8 citation statements)

References 24 publications

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“…Under the assumption that the sequence of probability densities 17) and Q(f, f ) is given by (1.10). For the investigation of equation (1.16) we refer to Wennberg, Wondmagegne [20] and Bagland [1].…”

Section: The Thermostatted Kac Master Equationmentioning

confidence: 99%

“…Let f be a given probability density on R with respect to the Lebesgue measure m. For each N ∈ N, let W N be a probability density on R N with respect to the product measure m (N ) . Then the sequence {W N } N ∈N of probability densities on R N is said to be f -chaotic if (1) Each W N is a symmetric function of the variables v 1 , v 2 , · · · , v N .…”

Section: Introductionmentioning

confidence: 99%

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Propagation of Chaos for the Thermostatted Kac Master Equation

2014

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The Kac model is a simplified model of an N -particle system in which the collisions of a real particle system are modeled by random jumps of pairs of particle velocities. Kac proved propagation of chaos for this model, and hence provided a rigorous validation of the corresponding Boltzmann equation. Starting with the same model we consider an N -particle system in which the particles are accelerated between the jumps by a constant uniform force field which conserves the total energy of the system. We show propagation of chaos for this model.

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Section: The Thermostatted Kac Master Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Propagation of Chaos for the Thermostatted Kac Master Equation

2014

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“…It is worth noting that to the best of our knowledge, this is the first time that this proof has been presented for the discrete thermostatted kinetic theory proposed in [13]. However similar investigations have been addressed for the thermostatted Kac equation [22][23][24][25]. Further applications can be envisaged for biosystems [26,27], in medicine [28] and for complex systems [29].…”

Section: Introductionmentioning

confidence: 91%

A Convergence Theorem for the Nonequilibrium States in the Discrete Thermostatted Kinetic Theory

Bianca

Menale

2019

Mathematics

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The existence and reaching of nonequilibrium stationary states are important issues that need to be taken into account in the development of mathematical modeling frameworks for far off equilibrium complex systems. The main result of this paper is the rigorous proof that the solution of the discrete thermostatted kinetic model catches the stationary solutions as time goes to infinity. The approach towards nonequilibrium stationary states is ensured by the presence of a dissipative term (thermostat) that counterbalances the action of an external force field. The main result is obtained by employing the Discrete Fourier Transform (DFT).

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“…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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Well-Posedness and Large Time Behaviour for the Non-cutoff Kac Equation with a Gaussian Thermostat

Cited by 14 publications

References 24 publications

Propagation of Chaos for the Thermostatted Kac Master Equation

Propagation of Chaos for the Thermostatted Kac Master Equation

A Convergence Theorem for the Nonequilibrium States in the Discrete Thermostatted Kinetic Theory

On the convergence toward nonequilibrium stationary states in thermostatted kinetic models

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