On the motion of a body in thermal equilibrium immersed in a perfect gas
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 2, p. 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 : https://doi.org/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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {2},
     year = {2008},
     pages = {263-275},
     doi = {10.1051/m2an:2008007},
     zbl = {1133.76046},
     mrnumber = {2405148},
     language = {en},
     url = {http://www.numdam.org/item/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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 2, pp. 263-275. doi : 10.1051/m2an:2008007. http://www.numdam.org/item/M2AN_2008__42_2_263_0/

[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 475547 | Zbl 1155.81383

[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 1909561 | Zbl 1031.82050

[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 2212220 | Zbl 1113.82059

[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 2353147 | Zbl pre05274854

[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 2361025 | Zbl 1134.76055

[6] R.L. Dobrushin, Vlasov equations. Sov. J. Funct. Anal. 13 (1979) 115-123. | MR 541637 | Zbl 0422.35068

[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 1805331 | Zbl 1127.82308

[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 740721 | Zbl 0575.76120

[10] H. Spohn, On the Vlasov hierarchy. Math. Meth. Appl. Sci. 3 (1981) 445-455. | MR 657065 | Zbl 0492.35067