Many-body aspects of approach to equilibrium
Journées équations aux dérivées partielles (2000), article no. 11, 12 p.

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 N 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.

@article{JEDP_2000____A11_0,
     author = {Carlen, Eric and Carvalho, M. C. and Loss, Michael},
     title = {Many-body aspects of approach to equilibrium},
     journal = {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://www.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://www.numdam.org/item/JEDP_2000____A11_0/
LA  - en
ID  - JEDP_2000____A11_0
ER  - 
%0 Journal Article
%A Carlen, Eric
%A Carvalho, M. C.
%A Loss, Michael
%T Many-body aspects of approach to equilibrium
%J Journées équations aux dérivées partielles
%D 2000
%P 1-12
%I Université de Nantes
%U http://www.numdam.org/item/JEDP_2000____A11_0/
%G en
%F JEDP_2000____A11_0
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://www.numdam.org/item/JEDP_2000____A11_0/

[1] Carlen, E., Carvalho, M. and Loss, M., (in preparation).

[2] Carlen, E., Gabetta, E. and Toscani, G., 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

[3] Diaconis, P. and Saloff-Coste, L., Bounds for Kac's Master equation, Commun. Math. Phys. 209, 729-755, 2000. | MR | Zbl

[4] Janvresse, E., Spectral Gap for Kac's model of Boltzmann Equation, Preprint 1999.

[5] Kac, M., 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] Gruenbaum, F. A., Propagation of chaos for the Boltzmann equation, Arch. Rational. Mech. Anal. 42, 323-345, 1971. | MR | Zbl

[7] Gruenbaum, F. A., Linearization for the Boltzmann equation, Trans. Amer. Math. Soc. 165, 425-449, 1972. | MR | Zbl

[8] Mckean, H., Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas, Arch. Rational Mech. Anal. 21, 343-367, 1966. | MR