Propagation of Chaos for the Thermostatted Kac Master Equation
- 126 Downloads
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.
KeywordsMaster equation Propagation of chaos Kinetic equation with a thermostat
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.
- 2.Bonetto, F., Carlen, E., Esposito, R., Lebowitz, J., Marra, R.: Propagation of chaos for a thermostated kinetic model. J. Statist. Phys. 154(1-2), 265–285 (2014)Google Scholar
- 4.Cercignani, C., Illner, R., and Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer, Berlin (1994)Google Scholar
- 6.Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkely Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, pp. 171–197. University of California Press, Berkerly (1956)Google Scholar
- 7.Lanford, O.: Time evolution of large classical system. In: Moser, E.J. (ed.) Lecture Notes in Physics, vol. 38, pp. 1–111. Springer, Berlin (1975)Google Scholar
- 8.Mishler, S., Mouhot, C.: Kac’s Program in Kinetic Theory. Springer, Berlin (2012)Google Scholar
- 9.Mishler, S., Mouhot, C., Wennberg, B.: A new approach to quantitative propagation of chaos for drift, diffusion and jump processes. Probab. Theory Relat. Fields. (2013). doi: 10.1007/s00440-013-0542-8
- 10.Sznitman, A.: Topics in propagation of chaos. Ecole d’Eté de Probabilités de Saint-Flour XIX-1989. In: Lecture Notes in Mathematics, vol. 1464, pp. 165–251. Springer, Berlin (1991)Google Scholar
- 11.Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander S., Serre D. (eds.) Handbook of Mathematical Fluid Dynamics. Elsevier Science, Amsterdam (2002)Google Scholar
- 13.Wondmagegne, Y.: Kinetic equations with a Gaussian thermostat. Doctoral Thesis, Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University (2005)Google Scholar