Uniform propagation of chaos for the thermostated Kac model

Cortez, Roberto; Tossounian, Hagop

doi:10.48550/arxiv.2001.08262

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2020

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“…We will refer to this rule as the weak thermostat. As in Kac's original model, propagation of chaos also holds in this case, as first shown in [6]; see also [16] for a quantitative result.…”

Section: Introductionsupporting

confidence: 60%

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On a thermostated Kac model with rescaling

Cortez,

Tossounian

2020

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We introduce a global thermostat on Kac's 1D model for the velocities of particles in a spacehomogeneous gas subjected to binary collisions, also interacting with a (local) Maxwellian thermostat. The global thermostat rescales the velocities of all the particles, thus restoring the total energy of the system, which leads to an additional drift term in the corresponding nonlinear kinetic equation. We prove ergodicity for this equation, and show that its equilibrium distribution has a density that, depending on the parameters of the model, can exhibit heavy tails, and whose behaviour at the origin can range from being analytic, to being C k , and even to blowing-up. Finally, we prove propagation of chaos for the associated N -particle system, with a uniform-in-time rate of order N −η in the squared 2-Wasserstein metric, for an explicit η ∈ (0, 1/3].

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Section: Introductionsupporting

confidence: 60%

“…The present paper can be seen as a continuation of our previous work [16], where we used similar tools to make quantitative the propagation of chaos result for the thermostated Kac model (without rescaling) in [6]. Here, the main objects of study are the thermostated finite particle system with rescaling and its corresponding kinetic equation (6).…”

Section: 2mentioning

confidence: 91%