Abstract
We study a system of M particles in contact with a large but finite reservoir of \({N \gg M}\) particles within the framework of the Kac master equation modeling random collisions. The reservoir is initially in equilibrium at temperature \({T=\beta^{-1}}\). We show that for large N, this evolution can be approximated by an effective equation in which the reservoir is described by a Maxwellian thermostat at temperature T. This approximation is proven for a suitable \({L^2}\) norm as well as for the Gabetta–Toscani–Wennberg (GTW) distance and is uniform in time.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bonetto F., Loss M., Vaidyanathan R.: The Kac model coupled to a thermostat. J. Stat. Phys. 156(4), 647–667 (2014)
Carlen, E., Carvalho, M.C., Loss, M.: Many-body aspects of approach to equilibrium. In: Journées “Équations aux Dérivées Partielles” (La Chapelle sur Erdre, 2000), pages Exp. No. XI, 12. Univ. Nantes, Nantes (2000)
Carlen E., Lebowitz J., Mouhot C.: Exponential approach to, and properties of, a non-equilibrium steady state in a dilute gas. Br. J. Probab. Stat. 29(2), 372–386 (2015)
Evans J.: Non-equilibrium steady states in kac’s model coupled to a thermostat. J. Stat. Phys. 164(5), 1103–1121 (2016)
Gabetta E., Toscani G., Wennberg B.: Metrics for probability distributions and the trend to equilibrium for solutions of the boltzmann equation. J. Stat. Phys. 81, 901–934 (1995)
Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, pp. 171–197. University of California Press, Berkeley (1956)
McKean H.P. Jr: Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas. Arch. Ration. Mech. Anal. 21, 343–367 (1966)
Tossounian, H., Vaidyanathan, R.: Partially thermostated Kac model. J. Math. Phys. 56(8) (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Mouhot
Rights and permissions
About this article
Cite this article
Bonetto, F., Loss, M., Tossounian, H. et al. Uniform Approximation of a Maxwellian Thermostat by Finite Reservoirs. Commun. Math. Phys. 351, 311–339 (2017). https://doi.org/10.1007/s00220-016-2803-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-016-2803-8