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The Coefficient of Restitution Does Not Exceed Unity

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Abstract

We study a classical mechanical problem in which a macroscopic ball is reflected by a non-deformable wall. The ball is modeled as a collection of classical particles bound together by an arbitrary potential, and its internal degrees of freedom are initially set to be in thermal equilibrium. The wall is represented by an arbitrary potential which is translation invariant in two directions. We then prove that the final normal momentum can exceed the initial normal momentum at most by O(■mkT), where m is the total mass of the ball, k the Boltzmann constant, and T the temperature. This implies the well-known statement in the title in the macroscopic limit where O(■mkT) is negligible. Our result may be interpreted as a rigorous demonstration of the second law of thermodynamics in a system where a macroscopic dynamics and microscopic degrees of freedom are intrinsically coupled with each other.

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References

  1. W. J. Stronge, Impact Mechanics. Cambridge University Press (2000).

  2. H. Hayakawa and H. Kuninaka, Phase Trans. 79:889 (2004),[cond-mat/0312005].

    Article  Google Scholar 

  3. M. Y. Louge and M. E. Adams, Phys. Rev. E65:021303 (2002).

    ADS  Google Scholar 

  4. H. Kuninaka and H. Hayakawa, Phys. Rev. Lett. 93:154301 (2004).

    Article  ADS  Google Scholar 

  5. A. Lenard, J. Stat. Phys. 19:575 (1978).

    Article  ADS  MathSciNet  Google Scholar 

  6. W. Pusz and S. L. Woronowicz, Commun. Math. Phys. 58:273 (1978).

    Article  ADS  MathSciNet  Google Scholar 

  7. W. Thirring, Quantum Mechanics of Large Systems. Springer (1983).

  8. C. Jarzynski, Phys. Rev. Lett. 78:2690 (1997).

    Article  ADS  Google Scholar 

  9. C. Jarzynski, J. Stat. Phys. 96:415 (1999).

    Article  MATH  Google Scholar 

  10. H. Tasaki, Statistical mechanical derivation of the second law of thermodynamics (unpublished note). [cond-mat/0009206].

  11. H. Tasaki, The second law of Thermodynamics as a theorem in quantum mechanics (unpublished note). [cond-mat/0011321].

  12. C. Maes and H. Tasaki, Second law of thermodynamics for macroscopic mechanics coupled to thermodynamic degrees of freedom. [cond-mat/0511419].

  13. S. Goldstein and J. L. Lebowitz, Physica D193 53 (2004). [cond-mat/0304251].

  14. R. L. Garrido, S. Goldstein and J. L. Lebowitz, Phys. Rev. Lett. 92:050602 (2004).

    Article  ADS  Google Scholar 

Download references

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  1. Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo, 171-8588, Japan

    Hal Tasaki

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  1. Hal Tasaki
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Correspondence to Hal Tasaki.

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Tasaki, H. The Coefficient of Restitution Does Not Exceed Unity. J Stat Phys 123, 1361–1374 (2006). https://doi.org/10.1007/s10955-006-9129-4

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  • DOI: https://doi.org/10.1007/s10955-006-9129-4

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