Figure 1.
Sampled and implied entropy decay for $ \alpha = 2 $ , and sampled entropy decay for $ \alpha = 0 $ , respectively
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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 |
In this paper, we proceed as suggested in the final section of [
Keywords:
Kinetic theory,
particle systems,
Kac process,
spectral gap,
functional analysis,
Monte Carlo simulations.
Mathematics Subject Classification:
Primary: 82C40, 39B42; Secondary: 82M31.
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. Barthe, D. 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. Carlen, M. 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. Carlen, J. 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. Carlen, J. 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. Carlen, E. 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. Nevai, T. 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. Barthe, D. 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. Carlen, M. 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. Carlen, J. 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. Carlen, J. 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. Carlen, E. 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. Nevai, T. 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 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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