On the motion of a body in thermal equilibrium immersed in a perfect gas
ESAIM: Modélisation mathématique et analyse numérique, Volume 42 (2008) no. 2, pp. 263-275.

We consider a body immersed in a perfect gas and moving under the action of a constant force. Body and gas are in thermal equilibrium. We assume a stochastic interaction body/medium: when a particle of the medium hits the body, it is absorbed and immediately re-emitted with a Maxwellian distribution. This system gives rise to a microscopic model of friction. We study the approach of the body velocity V(t) to the limiting velocity V and prove that, under suitable smallness assumptions, the approach to equilibrium is

|V(t)-V |C t d+1 ,
where d is the dimension of the space, and C is a positive constant. This approach is not exponential, as typical in friction problems, and even slower than for the same problem with elastic collisions.

DOI: 10.1051/m2an:2008007
Classification: 76P05, 82B40, 82C40, 35L45, 35L50
Keywords: kinetic theory of gases, Boltzmann equation, free molecular gas, friction problem, approach to equilibrium
@article{M2AN_2008__42_2_263_0,
     author = {Aoki, Kazuo and Cavallaro, Guido and Marchioro, Carlo and Pulvirenti, Mario},
     title = {On the motion of a body in thermal equilibrium immersed in a perfect gas},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {263--275},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {2},
     year = {2008},
     doi = {10.1051/m2an:2008007},
     mrnumber = {2405148},
     zbl = {1133.76046},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2008007/}
}
TY  - JOUR
AU  - Aoki, Kazuo
AU  - Cavallaro, Guido
AU  - Marchioro, Carlo
AU  - Pulvirenti, Mario
TI  - On the motion of a body in thermal equilibrium immersed in a perfect gas
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2008
SP  - 263
EP  - 275
VL  - 42
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2008007/
DO  - 10.1051/m2an:2008007
LA  - en
ID  - M2AN_2008__42_2_263_0
ER  - 
%0 Journal Article
%A Aoki, Kazuo
%A Cavallaro, Guido
%A Marchioro, Carlo
%A Pulvirenti, Mario
%T On the motion of a body in thermal equilibrium immersed in a perfect gas
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2008
%P 263-275
%V 42
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2008007/
%R 10.1051/m2an:2008007
%G en
%F M2AN_2008__42_2_263_0
Aoki, Kazuo; Cavallaro, Guido; Marchioro, Carlo; Pulvirenti, Mario. On the motion of a body in thermal equilibrium immersed in a perfect gas. ESAIM: Modélisation mathématique et analyse numérique, Volume 42 (2008) no. 2, pp. 263-275. doi : 10.1051/m2an:2008007. http://www.numdam.org/articles/10.1051/m2an:2008007/

[1] W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles. Comm. Math. Phys. 56 (1977) 101-113. | MR | Zbl

[2] P. Buttà, E. Caglioti and C. Marchioro, On the long time behavior of infinitely extended systems of particles interacting via Kac Potentials. J. Stat. Phys. 108 (2002) 317-339. | MR | Zbl

[3] S. Caprino, C. Marchioro and M. Pulvirenti, Approach to equilibrium in a microscopic model of friction. Comm. Math. Phys. 264 (2006) 167-189. | MR | Zbl

[4] S. Caprino, G. Cavallaro and C. Marchioro, On a microscopic model of viscous friction. Math. Models Methods Appl. Sci. 17 (2007) 1369-1403. | MR

[5] G. Cavallaro, On the motion of a convex body interacting with a perfect gas in the mean-field approximation. Rend. Mat. Appl. 27 (2007) 123-145. | MR | Zbl

[6] R.L. Dobrushin, Vlasov equations. Sov. J. Funct. Anal. 13 (1979) 115-123. | MR | Zbl

[7] C. Gruber and J. Piasecki, Stationary motion of the adiabatic piston. Physica A 268 (1999) 412-423.

[8] J.L. Lebowitz, J. Piasecki and Y. Sinai, Scaling dynamics of a massive piston in a ideal gas, in Hard Ball Systems and the Lorentz Gas, Encycl. Math. Sci. 101, Springer, Berlin (2000) 217-227. | MR | Zbl

[9] H. Neunzert, An Introduction to the Nonlinear Boltzmann-Vlasov Equation, in Kinetic Theories and the Boltzmann Equation, Montecatini (1981), Lecture Notes in Math. 1048, Springer, Berlin (1984) 60-110. | MR | Zbl

[10] H. Spohn, On the Vlasov hierarchy. Math. Meth. Appl. Sci. 3 (1981) 445-455. | MR | Zbl

Cited by Sources: