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A Counter Example to Cercignani’s Conjecture for the d Dimensional Kac Model


Kac’s d dimensional model gives a linear, many particle, binary collision model from which, under suitable conditions, the celebrated Boltzmann equation, in its spatially homogeneous form, arise as a mean field limit. The ergodicity of the evolution equation leads to questions about the relaxation rate, in hope that such a rate would pass on the Boltzmann equation as the number of particles goes to infinity. This program, starting with Kac and his one dimensional ‘Spectral Gap Conjecture’ at 1956, finally reached its conclusion in the Maxwellian case in a series of papers by authors such as Janvresse, Maslen, Carlen, Carvalho, Loss and Geronimo, but the hope to get a limiting relaxation rate for the Boltzmann equation with this linear method was already shown to be unrealistic (although the problem is still important and interesting due to its connection with the linearized Boltzmann operator). A less linear approach, via a many particle version of Cercignani’s conjecture, is the grounds for this paper. In our paper, we extend recent results by the author from the one dimensional Kac model to the d dimensional one, showing that the entropy-entropy production ratio, Γ N , still yields a very strong dependency in the number of particles of the problem when we consider the general case.

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  1. 1.

    Bobylev, A.V., Cercignani, C.: On the rate of entropy production for the Boltzmann equation. J. Stat. Phys. 94, 603–618 (1999)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  2. 2.

    Carlen, E.A., Carvalho, M.C., Loss, M.: Many body aspects of approach to equilibrium. In: Séminaire Equations aux Dérivées Partielles, La Chapelle sur Erdre, 2000. University of Nantes, Nantes (2000). Exp. No. XI, 12 pp

    Google Scholar 

  3. 3.

    Carlen, E.A., Carvalho, M.C., Le Roux, J., Loss, M., Villani, C.: Entropy and chaos in the Kac model. Kinet. Relat. Models 3, 85–122 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Carlen, E.A., Geronimo, J.S., Loss, M.: Determination of the spectral gap in the Kac model for physical momentum and energy-conserving collisions. SIAM J. Math. Anal. 40(1), 327–364 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Carrapatoso, K.: Quantitative and qualitative Kac’s chaos on the Boltzmann sphere. arXiv:1205.1241v1

  6. 6.

    Cercignani, C.: H-Theorem and trend to equilibrium in the kinetic theory of gasses. Arch. Mech. Stosow. 34(3), 231–241 (1982). 1983

    MathSciNet  ADS  MATH  Google Scholar 

  7. 7.

    Einav, A.: On Villani’s conjecture concerning entropy production for the Kac master equation. Kinet. Relat. Models 4(2), 479–497 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Janvresse, E.: Spectral gap for Kac’s model of Boltzmann equation. Ann. Probab. 29, 288–304 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, pp. 171–197. University of California Press, Berkeley, Los Angeles (1956)

    Google Scholar 

  10. 10.

    Lanford, O.E. III: Time evolution of large classical systems. Dynamical systems, theory and applications In: Recontres, Battelle Res. Inst., Seattle, Wash., 1974, Lecture Notes in Phys., vol. 38, pp. 1–111. Springer, Berlin (1975)

    Google Scholar 

  11. 11.

    Maslen, D.K.: The eigenvalues of Kac’s master equation. Math. Z. 243(2), 291–331 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    McKean, H.P. Jr.: An exponential formula for solving Boltzmann’s equation for a Maxwellian gas. J. Comb. Theory 2, 358–382 (1967)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Mischlet, S., Mouhot, C.: Kac’s program in kinetic theory. arXiv:1107.3251v1

  14. 14.

    Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Handbook of Mathematical Fluid Dynamics, vol. I, pp. 71–305. North-Holland, Amsterdam (2002)

    Chapter  Google Scholar 

  15. 15.

    Villani, C.: Cercignani’s conjecture is sometimes true and always almost true. Commun. Math. Phys. 234, 455–490 (2003)

    MathSciNet  ADS  MATH  Article  Google Scholar 

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The author would like to thank Clément Mouhot for many fruitful discussions and constant encouragement, as well as Kleber Carrapatoso for allowing him to read the preprint of his paper [5], helping to bridge the dimension gap.

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Correspondence to Amit Einav.

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The author was supported by ERC Grant MATKIT.

Appendix: A Fubini Type Theorem

Appendix: A Fubini Type Theorem

This appendix contains the proof to Theorem 2, which we felt would have encumbered the main article, but pose a necessary step in the proof of our main result.

Proof of Theorem 2

The proof relies heavily on the transformation (16) and the following Fubini-like formula for spheres (which can be found in [7]):

$$ \everymath{\displaystyle }\begin{array}[b]{@{}lll} \int _{\mathbb{S}^{m-1}(r)}fd\gamma^m_r &=& \frac{ \vert\mathbb{S}^{m-j-1} \vert}{ \vert\mathbb{S}^{m-1} \vert r^{m-2}} \int_{\sum_{i=1}^j x_i^2 \leq r^2}dx_1 \dots dx_j \Biggl(r^2-\sum_{i=1}^j x_i^2 \Biggr)^{\frac{m-j-2}{2}} \cr\noalign{\vspace{3pt}} &&{} \times \int_{\mathbb{S}^{m-j-1} (\sqrt {r^2-\sum _{i=1}^j x_i^2} )}fd \gamma^{m-j}_{\sqrt{r^2-\sum_{i=1}^j x_i^2}}, \end{array} $$

where \(d\gamma^{m}_{r}\) is the uniform probability measure on the appropriate sphere.

We start by defining the new variables

where R 1,R 2 are transformation like (16). We notice that under the above transformation the domain

$$\sum_{i=1}^N |v_i|^2=E, \qquad \sum_{i=1}^N v_i=z $$

transforms into

$$ \sum_{i=1, i\neq j}^{N-1} | \xi_i|^2+\frac{N}{N-j} \biggl(\xi_j- \frac{\sqrt{j}z}{N} \biggr)^2=E-\frac{|z|^2}{N} . $$

Denoting by \(\widetilde{\xi_{j}}=\sqrt{\frac{N}{N-j}} (\xi_{j}-\frac {\sqrt {j}z}{N} )\) and using the fact that R=R 1R 2 is orthogonal along with (61) we find that

and since

$$E-\frac{|z|^2}{N}-\sum_{i=1}^{j-1} | \xi_i|^2 -|\widetilde{\xi_j}|^2=E- \sum_{i=1}^j |v_i|^2 - \frac{|z-\sum_{i=1}^j v_i|^2}{N-j}, $$

the result follows. □

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Einav, A. A Counter Example to Cercignani’s Conjecture for the d Dimensional Kac Model. J Stat Phys 148, 1076–1103 (2012).

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  • Kac Model
  • Entropy production
  • Cercignani’s many particle conjecture