Journal of Statistical Physics

, Volume 146, Issue 3, pp 620–645 | Cite as

Decay of a Linear Pendulum in a Free-Molecular Gas and in a Special Lorentz Gas

  • Tetsuro TsujiEmail author
  • Kazuo Aoki


A circular disk without thickness is placed in a gas, and an external force, obeying Hooke’s law, is acting perpendicularly on the disk. If the disk is displaced perpendicularly from its equilibrium position and released, then it starts an oscillatory or non-oscillatory unsteady motion, which decays as time goes on because of the drag exerted by the gas molecules. This unsteady motion, i.e., the decay of this linear pendulum, is investigated numerically, under the diffuse reflection condition on the surface of the disk, with special interest in the manner of its decay, for two kinds of gases: one is a collisionless gas (or Knudsen gas) and the other is a special Lorentz gas interacting with a background. It is shown that the decay of the displacement of the disk is slow and is in proportion to an inverse power of time for the collisionless gas. The result complements the existing mathematical study of a similar problem (Caprino et al. in Math. Models Methods Appl. Sci. 17:1369–1403, 2007) in the case of non-oscillatory decay. It is also shown that the manner of the decay changes significantly for the special Lorentz gas.


Decay of pendulum Free-molecular gas Lorentz gas Kinetic theory of gases 


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  1. 1.
    Caprino, S., Marchioro, C., Pulvirenti, M.: Approach to equilibrium in a microscopic model of friction. Comm. Math. Phys. 264, 167–189 (2006) MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Caprino, S., Cavallaro, G., Marchioro, C.: On a microscopic model of viscous friction. Math. Models Methods Appl. Sci. 17, 1369–1403 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Cavallaro, G.: On the motion of a convex body interacting with a perfect gas in the mean-field approximation. Rend. Mat. Appl. (7) 27, 123–145 (2007) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Aoki, K., Cavallaro, G., Marchioro, C., Pulvirenti, M.: On the motion of a body in thermal equilibrium immersed in a perfect gas. Math. Model. Numer. Anal. 42, 263–275 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Aoki, K., Tsuji, T., Cavallaro, G.: Approach to steady motion of a plate moving in a free-molecular gas under a constant external force. Phys. Rev. E 80, 016309 (2009) ADSCrossRefGoogle Scholar
  6. 6.
    Tsuji, T., Aoki, K.: Decay of an oscillating plate in a free-molecular gas. In: Levin, D.A., Wysong, I.J., Garcia, A.L. (eds.) Rarefied Gas Dynamics. pp. 140–145. AIP, Melville (2011) Google Scholar
  7. 7.
    Sone, Y.: Molecular Gas Dynamics: Theory, Techniques, and Applications. Birkäuser, Boston (2007); Supplementary Notes and Errata: Kyoto University Research Information Repository. zbMATHGoogle Scholar
  8. 8.
    Cavallaro, G., Marchioro, C.: On the approach to equilibrium for a pendulum immersed in a Stokes fluid. Math. Models Methods Appl. Sci. 20, 1999–2019 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cavallaro, G., Marchioro, C., Tsuji, T.: Approach to equilibrium of a rotating sphere in a Stokes flow. Ann. Univ. Ferrara 57, 211–228 (2011) CrossRefGoogle Scholar
  10. 10.
    Gallavotti, G.: Statistical Mechanics: A Short Treatise. Springer, Berlin (1999) zbMATHGoogle Scholar
  11. 11.
    Caglioti, E., Golse, F.: On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas. J. Stat. Phys. 141, 264–317 (2010) MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Boltzmann, L.: Lectures on Gas Theory. Dover, New York (1995) Google Scholar
  13. 13.
    Takata, S.: Invitation to the kinetic theory. Nagare (J. Jpn. Soc. Fluid Mech.) 27, 387–396 (2008) (in Japanese) Google Scholar
  14. 14.
    Gruber, Ch., Piasecki, J.: Stationary motion of the adiabatic piston. Physica A 268, 412–423 (1999) CrossRefGoogle Scholar
  15. 15.
    Chernov, N., Lebowitz, J.L., Sinai, Ya.: Scaling dynamics of a massive piston in a cube filled with ideal gas: exact results. J. Stat. Phys. 109, 529–548 (2002) MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    Lebowitz, J.L., Piasecki, J., Sinai, Ya.: In: Hard Ball Systems and the Lorentz Gas. Encyclopedia of Mathematical Sciences, vol. 101, pp. 217–227. Springer, New York (2000) Google Scholar
  17. 17.
    Kestemont, E., Van den Broeck, C., Mansour, M.M.: The “adiabatic” piston: and yet it moves. Europhys. Lett. 49, 143–149 (2000) ADSCrossRefGoogle Scholar
  18. 18.
    Chernov, N., Lebowitz, J.L.: Dynamics of a massive piston in an ideal gas: oscillatory motion and approach to equilibrium. J. Stat. Phys. 109, 507–527 (2002) MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Caglioti, E., Chernov, N., Lebowitz, J.L.: Stability of solutions of hydrodynamic equations describing the scaling limit of a massive piston in an ideal gas. Nonlinearity 17, 897–923 (2004) MathSciNetADSzbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mechanical Engineering and ScienceKyoto UniversityKyotoJapan
  2. 2.Department of Mechanical Engineering and Science and Advanced Research Institute of Fluid Science and EngineeringKyoto UniversityKyotoJapan

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