Journal of Statistical Physics

, Volume 172, Issue 3, pp 795–823 | Cite as

Motion of a Rigid Body in a Special Lorentz Gas: Loss of Memory Effect

  • Kai KoikeEmail author


Linear motion of a rigid body in a special kind of Lorentz gas is mathematically analyzed. The rigid body moves against gas drag according to Newton’s equation. The gas model is a special Lorentz gas consisting of gas molecules and background obstacles, which was introduced in Tsuji and Aoki (J Stat Phys 146:620–645, 2012). The specular boundary condition is imposed on the resulting kinetic equation. This study complements the numerical study by Tsuji and Aoki cited above—although the setting in this paper is slightly different from theirs, qualitatively the same asymptotic behavior is proved: The velocity V(t) of the rigid body decays exponentially if the obstacles undergo thermal motion; if the obstacles are motionless, then the velocity V(t) decays algebraically with a rate \(t^{-\,5}\) independent of the spatial dimension. This demonstrates the idea that interaction of the molecules with the background obstacles destroys the memory effect due to recollision.


Lorentz gas Moving boundary problem Memory effect 



I thank T. Tsuji for his comments on the references.


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Authors and Affiliations

  1. 1.School of Fundamental Science and Technology, Graduate School of Science and Technology Keio UniversityKohokuku, YokohamaJapan
  2. 2.RIKEN Center for Advanced Intelligence ProjectTokyoJapan

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