doi: 10.3934/krm.2021045
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A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles

Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Campo Grande 016, 1749-016 Lisbon

*Corresponding author: Luís Simão Ferreira

Received  August 2021 Revised  November 2021 Early access December 2021

Fund Project: Supported by CMAF-CIO grant UIDP/04561/2020

In this paper, we proceed as suggested in the final section of [2] and prove a lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. This bound turns out to be around $ 0.02 $, which is already physically meaningful, and we perform Monte Carlo simulations to provide a better empirical estimate for this value via entropy production inequalities. This finishes a complete quantitative estimate of the spectral gap of the Kac process.

Citation: Luís Simão Ferreira. A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. Kinetic & Related Models, doi: 10.3934/krm.2021045
References:
[1]

F. BartheD. Cordero-Erausquin and B. Maurey, Entropy of spherical marginals and related inequalities, J. Math. Pures Appl. (9), 86 (2006), 89-99.  doi: 10.1016/j.matpur.2006.04.003.  Google Scholar

[2]

E. A. CarlenM. C. Carvalho and M. Loss, Spectral gaps for reversible Markov processes with chaotic invariant measures: The Kac process with hard sphere collisions in three dimensions, Ann. Probab., 48 (2020), 2807-2844.  doi: 10.1214/20-AOP1437.  Google Scholar

[3]

E. A. CarlenJ. S. Geronimo and M. Loss, Determination of the spectral gap in the Kac model for physical momentum and energy conserving collisions, SIAM J. Math. Anal., 40 (2008), 327-364.  doi: 10.1137/070695423.  Google Scholar

[4]

E. A. CarlenJ. S. Geronimo and M. Loss, On the Markov sequence problem for Jacobi polynomials, Adv. Math., 226 (2011), 3426-3466.  doi: 10.1016/j.aim.2010.10.024.  Google Scholar

[5]

E. A. CarlenE. H. Lieb and M. Loss, A sharp analog of Young's inequality on $S^N$ and related entropy inequalities, J. Geom. Anal., 14 (2004), 487-520.  doi: 10.1007/BF02922101.  Google Scholar

[6]

E. Janvresse, Spectral gap for Kac's model of Boltzmann equation, Ann. Probab., 29 (2001), 288-304.  doi: 10.1214/aop/1008956330.  Google Scholar

[7]

M. Kac, Foundations of kinetic theory, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, University of California Press, Berkeley and Los Angeles, Calif., 1956,171–197.  Google Scholar

[8]

P. NevaiT. Erdélyi and A. P. Magnus, Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal., 25 (1994), 602-614.  doi: 10.1137/S0036141092236863.  Google Scholar

[9]

G. Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, 23, AMS, Providence R.I., 1939. doi: 10.1090/coll/023.  Google Scholar

[10]

C. Villani, Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys., 234 (2003), 455-490.  doi: 10.1007/s00220-002-0777-1.  Google Scholar

show all references

References:
[1]

F. BartheD. Cordero-Erausquin and B. Maurey, Entropy of spherical marginals and related inequalities, J. Math. Pures Appl. (9), 86 (2006), 89-99.  doi: 10.1016/j.matpur.2006.04.003.  Google Scholar

[2]

E. A. CarlenM. C. Carvalho and M. Loss, Spectral gaps for reversible Markov processes with chaotic invariant measures: The Kac process with hard sphere collisions in three dimensions, Ann. Probab., 48 (2020), 2807-2844.  doi: 10.1214/20-AOP1437.  Google Scholar

[3]

E. A. CarlenJ. S. Geronimo and M. Loss, Determination of the spectral gap in the Kac model for physical momentum and energy conserving collisions, SIAM J. Math. Anal., 40 (2008), 327-364.  doi: 10.1137/070695423.  Google Scholar

[4]

E. A. CarlenJ. S. Geronimo and M. Loss, On the Markov sequence problem for Jacobi polynomials, Adv. Math., 226 (2011), 3426-3466.  doi: 10.1016/j.aim.2010.10.024.  Google Scholar

[5]

E. A. CarlenE. H. Lieb and M. Loss, A sharp analog of Young's inequality on $S^N$ and related entropy inequalities, J. Geom. Anal., 14 (2004), 487-520.  doi: 10.1007/BF02922101.  Google Scholar

[6]

E. Janvresse, Spectral gap for Kac's model of Boltzmann equation, Ann. Probab., 29 (2001), 288-304.  doi: 10.1214/aop/1008956330.  Google Scholar

[7]

M. Kac, Foundations of kinetic theory, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, University of California Press, Berkeley and Los Angeles, Calif., 1956,171–197.  Google Scholar

[8]

P. NevaiT. Erdélyi and A. P. Magnus, Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal., 25 (1994), 602-614.  doi: 10.1137/S0036141092236863.  Google Scholar

[9]

G. Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, 23, AMS, Providence R.I., 1939. doi: 10.1090/coll/023.  Google Scholar

[10]

C. Villani, Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys., 234 (2003), 455-490.  doi: 10.1007/s00220-002-0777-1.  Google Scholar

Figure 1.  Sampled and implied entropy decay for $ \alpha = 2 $, and sampled entropy decay for $ \alpha = 0 $, respectively
Figure 2.  $ \alpha = 0 $, sampled velocity
Figure 3.  $ \alpha = 2 $, sampled velocity
Figure 4.  $ \alpha = 2 $, implied velocity
Figure 5.  First 300 values of $ \kappa_{n, \ell} $, for $ \ell = 5, 15 $, and overlap of the plots from $ \ell = 6 $ to $ 10 $
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