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Many-body aspects of approach to equilibrium. (English) Zbl 1213.81230

Journées “Équations aux dérivées partielles”, La Chapelle sur Erdre, Nantes, France, 5 au 9 juin 2000. Exposés Nos. I-XX. Nantes: Université de Nantes (ISBN 2-86939-157-9/pbk). Exp. No. 11, 12 p. (2000).
Summary: 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.
For the entire collection see [Zbl 0990.00045].

MSC:

81V70 Many-body theory; quantum Hall effect
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
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