Propagation of Chaos for the Thermostatted Kac Master Equation

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• 作者：Eric Carlen (1)
  Dawan Mustafa (2) (3)
  Bernt Wennberg (2) (3)
  
  1. Hill Center ; Department of Mathematics ; Rutgers University ; 110 Frelinghuysen Road ; Piscataway ; NJ ; 08854-8019 ; USA
  2. Department of Mathematical Sciences ; Chalmers University of Technology ; Gothenburg ; Sweden
  3. Department of Mathematical Sciences ; University of Gothenburg ; 41296 ; Gothenburg ; Sweden
  
• 关键词：Master equation ; Propagation of chaos ; Kinetic equation with a thermostat
• 刊名：Journal of Statistical Physics
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：158
• 期：6
• 页码：1341-1378
• 全文大小：437 KB
• 参考文献：1. Bagland, V (2010) Well-posedness and large time behaviour for the non-cutoff Kac equation with a Gaussian thermostat. J. Statist. Phys. 138: pp. 838-875 CrossRef
  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鈥?85 (2014)
  3. Carlen, E, Degond, P, Wennberg, B (2013) Kinetic limits for pair-interaction driven master equations and biological swarm models. Math. Models Methods Appl. Sci. 23: pp. 1339-1376 CrossRef
  4. Cercignani, C., Illner, R., and Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer, Berlin (1994)
  5. Gr眉nbaum, A (1971) Propgation of chaos for the Boltzmann equation. Arch. Ration. Mech. Anal. 42: pp. 323-345 CrossRef
  6. Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkely Symposium on Mathematical Statistics and Probability, 1954鈥?955, vol. III, pp. 171鈥?97. University of California Press, Berkerly (1956)
  7. Lanford, O.: Time evolution of large classical system. In: Moser, E.J. (ed.) Lecture Notes in Physics, vol. 38, pp. 1鈥?11. Springer, Berlin (1975)
  8. Mishler, S., Mouhot, C.: Kac鈥檚 Program in Kinetic Theory. Springer, Berlin (2012)
  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鈥橢t茅 de Probabilit茅s de Saint-Flour XIX-1989. In: Lecture Notes in Mathematics, vol. 1464, pp. 165鈥?51. Springer, Berlin (1991)
  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)
  12. Wennberg, B, Wondmagegne, Y (2006) The Kac equation with a thermostatted force field. J. Statist. Phys. 124: pp. 859-880 CrossRef
  13. Wondmagegne, Y.: Kinetic equations with a Gaussian thermostat. Doctoral Thesis, Department of Mathematical Sciences, Chalmers University of Technology and G枚teborg University (2005)
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Statistical Physics
  Mathematical and Computational Physics
  Physical Chemistry
  Quantum Physics
  
• 出版者：Springer Netherlands
• ISSN：1572-9613

文摘

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.