February  2022, 15(1): 91-117. doi: 10.3934/krm.2021045

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 Published  February 2022 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 and Related Models, 2022, 15 (1) : 91-117. 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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[9]

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

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[9]

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

[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.

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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