Many-body aspects of approach to equilibrium

Eric Carlen; M. C. Carvalho; Michael Loss

Journées équations aux dérivées partielles (2000)

  • page 1-12
  • ISSN: 0752-0360

Abstract

top
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.

How to cite

top

Carlen, Eric, Carvalho, M. C., and Loss, Michael. "Many-body aspects of approach to equilibrium." Journées équations aux dérivées partielles (2000): 1-12. <http://eudml.org/doc/93388>.

@article{Carlen2000,
abstract = {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.},
author = {Carlen, Eric, Carvalho, M. C., Loss, Michael},
journal = {Journées équations aux dérivées partielles},
language = {eng},
pages = {1-12},
publisher = {Université de Nantes},
title = {Many-body aspects of approach to equilibrium},
url = {http://eudml.org/doc/93388},
year = {2000},
}

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
PB - Université de Nantes
SP - 1
EP - 12
AB - 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.
LA - eng
UR - http://eudml.org/doc/93388
ER -

References

top
  1. [1] Carlen, E., Carvalho, M. and Loss, M., (in preparation). 
  2. [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. Zbl0927.76088MR99k:82060
  3. [3] Diaconis, P. and Saloff-Coste, L., Bounds for Kac's Master equation, Commun. Math. Phys. 209, 729-755, 2000. Zbl0953.60098MR2002e:60107
  4. [4] Janvresse, E., Spectral Gap for Kac's model of Boltzmann Equation, Preprint 1999. Zbl1034.82049
  5. [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. Zbl0072.42802MR18,960i
  6. [6] Gruenbaum, F. A., Propagation of chaos for the Boltzmann equation, Arch. Rational. Mech. Anal. 42, 323-345, 1971. Zbl0236.45011MR48 #13106
  7. [7] Gruenbaum, F. A., Linearization for the Boltzmann equation, Trans. Amer. Math. Soc. 165, 425-449, 1972. Zbl0236.45012MR45 #4783
  8. [8] McKean, H., Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas, Arch. Rational Mech. Anal. 21, 343-367, 1966. Zbl1302.60049MR35 #4963

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.