Kinetic theory and approach to equilibrium is usually studied in the realm of the Boltzmann equation. With a few notable exceptions not much is known about the solutions of this equation and about its derivation from fundamental principles. In 1956 Mark Kac introduced a probabilistic model of interacting particles. The velocity distribution is governed by a Markov semi group and the evolution of its single particle marginals is governed (in the infinite particle limit) by a caricature of the spatially homogeneous Boltzmann equation. In joint work with Eric Carlen and Maria Carvalho we compute the gap of the generator of this Markov semigroup and show that the best possible rate of approach to equilibrium in the Kac model is precisely the one predicted by the linearized Boltzmann equation. Similar, but less precise results hold for maxwellian molecules.
@incollection{JEDP_2000____A11_0, author = {Carlen, Eric and Carvalho, M. C. and Loss, Michael}, title = {Many-body aspects of approach to equilibrium}, booktitle = {}, series = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, eid = {11}, pages = {1--12}, publisher = {Universit\'e de Nantes}, year = {2000}, mrnumber = {1775687}, zbl = {01808701}, language = {en}, url = {http://archive.numdam.org/item/JEDP_2000____A11_0/} }
TY - JOUR AU - Carlen, Eric AU - Carvalho, M. C. AU - Loss, Michael TI - Many-body aspects of approach to equilibrium JO - Journées équations aux dérivées partielles PY - 2000 SP - 1 EP - 12 PB - Université de Nantes UR - http://archive.numdam.org/item/JEDP_2000____A11_0/ LA - en ID - JEDP_2000____A11_0 ER -
Carlen, Eric; Carvalho, M. C.; Loss, Michael. Many-body aspects of approach to equilibrium. Journées équations aux dérivées partielles (2000), article no. 11, 12 p. http://archive.numdam.org/item/JEDP_2000____A11_0/
[1]
, and , (in preparation).[2] Propagation of Smoothness and the Rate of Exponential Convergence to Equilibrium for a Spatially Homogeneous Maxwellian Gas, Commun. Math. Phys. 205, 521-546, 1999. | MR | Zbl
, and ,[3] Bounds for Kac's Master equation, Commun. Math. Phys. 209, 729-755, 2000. | MR | Zbl
and ,[4] Spectral Gap for Kac's model of Boltzmann Equation, Preprint 1999.
,[5] Foundations of kinetic theory, Proc. 3rd Berkeley symp. Math. Stat. Prob., J. Neyman, ed. Univ. of California, vol 3, pp. 171-197, 1956. | MR | Zbl
,[6] Propagation of chaos for the Boltzmann equation, Arch. Rational. Mech. Anal. 42, 323-345, 1971. | MR | Zbl
,[7] Linearization for the Boltzmann equation, Trans. Amer. Math. Soc. 165, 425-449, 1972. | MR | Zbl
,[8] Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas, Arch. Rational Mech. Anal. 21, 343-367, 1966. | MR
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