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April  2018, 11(2): 219-238. doi: 10.3934/krm.2018012

Entropy production inequalities for the Kac Walk

Eric A. Carlen 1, , Maria C. Carvalho 2, and  Amit Einav 3,

1. 

Department of Mathematics, Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway NJ 08854-8019, USA
2. 

CMAF-CIO, University of Lisbon, P 1749-016 Lisbon, Portugal
3. 

Departments of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

Received  April 2017 Published  January 2018

Fund Project: 1Work partially supported by U.S. National Science Foundation grant DMS 1501007. 2Work partially supported by supported by Fundação para a Ciência e Tecnologia (PTDC/MAT/100983/2008, PEst-OE/MAT/UI0209/2013, UID/MAT/04561/2013). 3Work supported by EPSRC grant EP/L002302/1.

Mark Kac introduced what is now called 'the Kac Walk' with the aim of investigating the spatially homogeneous Boltzmann equation by probabilistic means. Much recent work, discussed below, on Kac's program has run in the other direction: using recent results on the Boltzmann equation, or its one-dimensional analog, the non-linear Kac-Boltzmann equation, to prove results for the Kac Walk. Here we investigate new functional inequalities for the Kac Walk pertaining to entropy production, and introduce a new form of 'chaoticity'. We then show how these entropy production inequalities imply entropy production inequalities for the Kac-Boltzmann equation. This results validate Kac's program for proving results on the non-linear Boltzmann equation via analysis of the Kac Walk, and they constitute a partial solution to the 'Almost' Cercignani Conjecture on the sphere.

Keywords: Kac Walk, chaoticity, entropy method, entropy production, cercignani conjecture.
Mathematics Subject Classification: Primary: 81V99, 82B10, 94A17.
Citation: Eric A. Carlen, Maria C. Carvalho, Amit Einav. Entropy production inequalities for the Kac Walk. Kinetic & Related Models, 2018, 11 (2) : 219-238. doi: 10.3934/krm.2018012
References:
[1]

A. V. Bobylev and C. Cercignani, On the rate of entropy production for the Boltzmann equation, J. Stat. Phys., 94 (1999), 603-618.  doi: 10.1023/A:1004537522686.  Google Scholar

[2]

T. Carleman, Sur la théorie de l'equation intégrodifférentielle de Boltzmann, Acta Math., 60 (1933), 91-146.   Google Scholar

[3]

E. A. Carlen, M. C. Carvalho and M. Loss, Many body aspects of approach to equilibrium, Journées "Équations aux dérivées partielles", Ⅺ (2000), 12pp.  Google Scholar

[4]

E. A. Carlen, M. C. Carvalho and M. Loss, Determination of the spectral gap for Kac's master equation and related stochastic evolution, Acta Mathematica, 191 (2003), 1-54.  doi: 10.1007/BF02392695.  Google Scholar

[5]

E. A. Carlen, M. C. Carvalho and M. Loss, Spectral gap for the Kac model with hard sphere collisions, J. Func. Anal., 266 (2014), 1787-1832.  doi: 10.1016/j.jfa.2013.08.024.  Google Scholar

[6]

E. A. Carlen, M. C. Carvalho, J. Le Roux, M. Loss and C. Villani, Entropy and chaos in the Kac model, Kinet. Relat. Models, 3 (2010), 85-122.  doi: 10.3934/krm.2010.3.85.  Google Scholar

[7]

E. A. Carlen, J. 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

[8]

K. Carrapatoso, Quantitative and qualitative Kac's chaos on the Boltzmann sphere, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), 993-1039.  doi: 10.1214/14-AIHP612.  Google Scholar

[9]

K. Carrapatoso and A. EInav, Chaos and entropic chaos in Kac's model without high moments, Electron. J. Probab., 18 (2013), 1-38.   Google Scholar

[10]

C. Cercignani, $ H$-theorem and trend to equilibrium in the kinetic theory of gases, Arch. Mech., 34 (1982), 231-241.   Google Scholar

[11]

I. Csiszár, Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hungar., 2 (1967), 299-318.   Google Scholar

[12]

L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann and Kac equations, Arch. Rational Mech. Anal., 123 (1993), 387-404.  doi: 10.1007/BF00375586.  Google Scholar

[13]

P. Diaconis and L. Saloff-Coste, Bounds for Kac's master equation, Comm. Math. Phys., 209 (2000), 729-755.  doi: 10.1007/s002200050036.  Google Scholar

[14]

A. Einav, On Villani's conjecture concerning entropy production for the Kac master equation, Kinet. Relat. Models, 4 (2011), 479-497.  doi: 10.3934/krm.2011.4.479.  Google Scholar

[15]

A. Einav, A counter example to Cercignani's conjecture for the $ d-$-dimensional Kac model, J. Stat. Phys., 148 (2012), 1076-1103.  doi: 10.1007/s10955-012-0565-z.  Google Scholar

[16]

A. Einav A, A few ways to destroy entropic chaoticity on Kac's sphere, Commun. Math. Sci., 12 (2014), 41-60.  doi: 10.4310/CMS.2014.v12.n1.a3.  Google Scholar

[17]

F. A. Grünbaum, Propagation of chaos for the Boltzmann equation, Arch. Rational Mech. Anal., 42 (1971), 323-345.   Google Scholar

[18]

M. Hauray and M. Mischler, On Kac's chaos and related problems, J. Funct. Anal., 266 (2014), 6055-6157.  doi: 10.1016/j.jfa.2014.02.030.  Google Scholar

[19]

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

[20]

M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 3 (1956), 171-197.   Google Scholar

[21]

M. Kac, Probability and Related Topics in Physical Sciences, Wiley Interscience Publ. LTD., New York, 1959. doi: 10. 1063/1. 3056918.  Google Scholar

[22]

S. Kullback, A lower bound for discrimination information in terms of variation, IEEE Transa. Information Theory, 13 (1967), 126-127.  doi: 10.1109/TIT.1967.1053968.  Google Scholar

[23]

H. McKean, Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas, Arch. Rational Mech. Anal., 21 (1966), 343-367.  doi: 10.1007/BF00264463.  Google Scholar

[24]

S. Mischler and C. Mouhot, Kac's program in kinetic theory, Invent. Math., bf 193 (2013), 1-147.  doi: 10.1007/s00222-012-0422-3.  Google Scholar

[25]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, 1963.  Google Scholar

[26]

M. Rousset, A $ N$-uniform quantitative Tanaka's theorem for the conservative Kac; s $ N$-particle system with Maxwell molecules, preprint, arXiv: 1407.1965. Google Scholar

[27]

A. S. Sznitman, Équations de type de Boltzmann, spatialement homogènes, Z. Wahrsch. Verw. Gebiete, 66 (1984), 559-592.  doi: 10.1007/BF00531891.  Google Scholar

[28]

A. S. Sznitman, Topics in propagation of chaos, In École dÉté de Probabilités de Saint-Flour XIX, 1989, Lecture Notes in Math. 1464 (1991), Springer, Berlin, 165-251.  Google Scholar

[29]

H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Gebiete, 46 (1978/79), 67-105.  doi: 10.1007/BF00535689.  Google Scholar

[30]

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]

A. V. Bobylev and C. Cercignani, On the rate of entropy production for the Boltzmann equation, J. Stat. Phys., 94 (1999), 603-618.  doi: 10.1023/A:1004537522686.  Google Scholar

[2]

T. Carleman, Sur la théorie de l'equation intégrodifférentielle de Boltzmann, Acta Math., 60 (1933), 91-146.   Google Scholar

[3]

E. A. Carlen, M. C. Carvalho and M. Loss, Many body aspects of approach to equilibrium, Journées "Équations aux dérivées partielles", Ⅺ (2000), 12pp.  Google Scholar

[4]

E. A. Carlen, M. C. Carvalho and M. Loss, Determination of the spectral gap for Kac's master equation and related stochastic evolution, Acta Mathematica, 191 (2003), 1-54.  doi: 10.1007/BF02392695.  Google Scholar

[5]

E. A. Carlen, M. C. Carvalho and M. Loss, Spectral gap for the Kac model with hard sphere collisions, J. Func. Anal., 266 (2014), 1787-1832.  doi: 10.1016/j.jfa.2013.08.024.  Google Scholar

[6]

E. A. Carlen, M. C. Carvalho, J. Le Roux, M. Loss and C. Villani, Entropy and chaos in the Kac model, Kinet. Relat. Models, 3 (2010), 85-122.  doi: 10.3934/krm.2010.3.85.  Google Scholar

[7]

E. A. Carlen, J. 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

[8]

K. Carrapatoso, Quantitative and qualitative Kac's chaos on the Boltzmann sphere, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), 993-1039.  doi: 10.1214/14-AIHP612.  Google Scholar

[9]

K. Carrapatoso and A. EInav, Chaos and entropic chaos in Kac's model without high moments, Electron. J. Probab., 18 (2013), 1-38.   Google Scholar

[10]

C. Cercignani, $ H$-theorem and trend to equilibrium in the kinetic theory of gases, Arch. Mech., 34 (1982), 231-241.   Google Scholar

[11]

I. Csiszár, Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hungar., 2 (1967), 299-318.   Google Scholar

[12]

L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann and Kac equations, Arch. Rational Mech. Anal., 123 (1993), 387-404.  doi: 10.1007/BF00375586.  Google Scholar

[13]

P. Diaconis and L. Saloff-Coste, Bounds for Kac's master equation, Comm. Math. Phys., 209 (2000), 729-755.  doi: 10.1007/s002200050036.  Google Scholar

[14]

A. Einav, On Villani's conjecture concerning entropy production for the Kac master equation, Kinet. Relat. Models, 4 (2011), 479-497.  doi: 10.3934/krm.2011.4.479.  Google Scholar

[15]

A. Einav, A counter example to Cercignani's conjecture for the $ d-$-dimensional Kac model, J. Stat. Phys., 148 (2012), 1076-1103.  doi: 10.1007/s10955-012-0565-z.  Google Scholar

[16]

A. Einav A, A few ways to destroy entropic chaoticity on Kac's sphere, Commun. Math. Sci., 12 (2014), 41-60.  doi: 10.4310/CMS.2014.v12.n1.a3.  Google Scholar

[17]

F. A. Grünbaum, Propagation of chaos for the Boltzmann equation, Arch. Rational Mech. Anal., 42 (1971), 323-345.   Google Scholar

[18]

M. Hauray and M. Mischler, On Kac's chaos and related problems, J. Funct. Anal., 266 (2014), 6055-6157.  doi: 10.1016/j.jfa.2014.02.030.  Google Scholar

[19]

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

[20]

M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 3 (1956), 171-197.   Google Scholar

[21]

M. Kac, Probability and Related Topics in Physical Sciences, Wiley Interscience Publ. LTD., New York, 1959. doi: 10. 1063/1. 3056918.  Google Scholar

[22]

S. Kullback, A lower bound for discrimination information in terms of variation, IEEE Transa. Information Theory, 13 (1967), 126-127.  doi: 10.1109/TIT.1967.1053968.  Google Scholar

[23]

H. McKean, Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas, Arch. Rational Mech. Anal., 21 (1966), 343-367.  doi: 10.1007/BF00264463.  Google Scholar

[24]

S. Mischler and C. Mouhot, Kac's program in kinetic theory, Invent. Math., bf 193 (2013), 1-147.  doi: 10.1007/s00222-012-0422-3.  Google Scholar

[25]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, 1963.  Google Scholar

[26]

M. Rousset, A $ N$-uniform quantitative Tanaka's theorem for the conservative Kac; s $ N$-particle system with Maxwell molecules, preprint, arXiv: 1407.1965. Google Scholar

[27]

A. S. Sznitman, Équations de type de Boltzmann, spatialement homogènes, Z. Wahrsch. Verw. Gebiete, 66 (1984), 559-592.  doi: 10.1007/BF00531891.  Google Scholar

[28]

A. S. Sznitman, Topics in propagation of chaos, In École dÉté de Probabilités de Saint-Flour XIX, 1989, Lecture Notes in Math. 1464 (1991), Springer, Berlin, 165-251.  Google Scholar

[29]

H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Gebiete, 46 (1978/79), 67-105.  doi: 10.1007/BF00535689.  Google Scholar

[30]

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

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