On the motion of a body in thermal equilibrium immersed in a perfect gas

Kazuo Aoki; Guido Cavallaro; Carlo Marchioro; Mario Pulvirenti

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 2, page 263-275
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Aoki, Kazuo, et al. "On the motion of a body in thermal equilibrium immersed in a perfect gas." ESAIM: Mathematical Modelling and Numerical Analysis 42.2 (2008): 263-275. <http://eudml.org/doc/250378>.

@article{Aoki2008,
abstract = { 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 $V_\infty$ and prove that, under suitable smallness assumptions, the approach to equilibrium is $$ |V(t)-V\_\infty|\approx \frac\{C\}\{t^\{d+1\}\}, $$ 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. },
author = {Aoki, Kazuo, Cavallaro, Guido, Marchioro, Carlo, Pulvirenti, Mario},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Kinetic theory of gases; Boltzmann equation; free molecular gas; friction problem; approach to equilibrium.; approach to equilibrium},
language = {eng},
month = {3},
number = {2},
pages = {263-275},
publisher = {EDP Sciences},
title = {On the motion of a body in thermal equilibrium immersed in a perfect gas},
url = {http://eudml.org/doc/250378},
volume = {42},
year = {2008},
}

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: Mathematical Modelling and Numerical Analysis
DA - 2008/3//
PB - EDP Sciences
VL - 42
IS - 2
SP - 263
EP - 275
AB - 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 $V_\infty$ and prove that, under suitable smallness assumptions, the approach to equilibrium is $$ |V(t)-V_\infty|\approx \frac{C}{t^{d+1}}, $$ 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.
LA - eng
KW - Kinetic theory of gases; Boltzmann equation; free molecular gas; friction problem; approach to equilibrium.; approach to equilibrium
UR - http://eudml.org/doc/250378
ER -

References

top
  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.  Zbl1155.81383
  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.  Zbl1031.82050
  3. S. Caprino, C. Marchioro and M. Pulvirenti, Approach to equilibrium in a microscopic model of friction. Comm. Math. Phys.264 (2006) 167–189.  Zbl1113.82059
  4. S. Caprino, G. Cavallaro and C. Marchioro, On a microscopic model of viscous friction. Math. Models Methods Appl. Sci.17 (2007) 1369–1403.  Zbl1216.70008
  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.  Zbl1134.76055
  6. R.L. Dobrushin, Vlasov equations. Sov. J. Funct. Anal.13 (1979) 115–123.  Zbl0422.35068
  7. C. Gruber and J. Piasecki, Stationary motion of the adiabatic piston. Physica A268 (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.  Zbl1127.82308
  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.  
  10. H. Spohn, On the Vlasov hierarchy. Math. Meth. Appl. Sci.3 (1981) 445–455.  Zbl0492.35067

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.