Abstract
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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Notes
This is a consequence of the first inequality.
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Acknowledgments
We would like to thank an anonymous referee for having read the previous version of this paper very carefully and for pointing out several important issues. E.C. would like to thank Chalmers Institute of Technology during a visit in the Spring of 2012, and would like to acknowledge support from N.S.F. Grant DMS-1201354. D.M. and B.W. would like to thank Kleber Carrapatoso for the discussions during the initial phase of this work. This work was supported by Grants from the Swedish Science Council and the Knut and Alice Wallenberg foundation.
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Carlen, E., Mustafa, D. & Wennberg, B. Propagation of Chaos for the Thermostatted Kac Master Equation. J Stat Phys 158, 1341–1378 (2015). https://doi.org/10.1007/s10955-014-1155-z
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DOI: https://doi.org/10.1007/s10955-014-1155-z