×

On Kac’s chaos and related problems. (English) Zbl 1396.60102

Summary: This paper is devoted to establish quantitative and qualitative estimates related to the notion of chaos as firstly formulated by M. Kac in his study of mean-field limit for systems of \(N\) undistinguishable particles as \(N\to\infty\) [in: Proc. 3rd Berkeley Symp. Math. Stat. Probab. 3, 171–197 (1956; Zbl 0072.42802)]. First, we quantitatively liken three usual measures of Kac’s chaos, some involving all the \(N\) variables, others involving a finite fixed number of variables. Next, we define the notion of entropy chaos and Fisher information chaos in a similar way as defined by E. A. Carlen et al. [Kinet. Relat. Models 3, No. 1, 85–122 (2010; Zbl 1186.76675)]. We show that Fisher information chaos is stronger than entropy chaos, which in turn is stronger than Kac’s chaos. We also establish that Kac’s chaos plus Fisher information bound implies entropy chaos. We then extend our analysis to the framework of probability measures with support on the Kac’s spheres, revisiting Carlen et al. [loc.cit.] and giving a possible answer to [loc.cit., Open problem 11]. Last, we consider the context of probability measures mixtures introduced by B. de Finetti [Ann. Inst. Henri Poincaré 7, 1–68 (1937; Zbl 0017.07602)], E. Hewitt and L. J. Savage [Trans. Am. Math. Soc. 80, 470–501 (1955; Zbl 0066.29604)]. We define the (level 3) Fisher information for mixtures and prove that it is l.s.c.and affine, as that was done in [D. W. Robinson and D. Ruelle, Commun. Math. Phys. 5, 288–300 (1967; Zbl 0144.48205)] for the level 3 Boltzmann’s entropy.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
37B40 Topological entropy
60A10 Probabilistic measure theory
60B10 Convergence of probability measures
60G09 Exchangeability for stochastic processes
82C40 Kinetic theory of gases in time-dependent statistical mechanics
94A17 Measures of information, entropy
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ajtai, M.; Komlós, J.; Tusnády, G., On optimal matchings, Combinatorica, 4, 4, 259-264 (1984) · Zbl 0562.60012
[2] Arkeryd, L.; Caprino, S.; Ianiro, N., The homogeneous Boltzmann hierarchy and statistical solutions to the homogeneous Boltzmann equation, J. Stat. Phys., 63, 1-2, 345-361 (1991)
[3] Artstein, S.; Ball, K. M.; Barthe, F.; Naor, A., On the rate of convergence in the entropic central limit theorem, Probab. Theory Related Fields, 129, 3, 381-390 (2004) · Zbl 1055.94004
[4] Barthe, F.; Bordenave, C., Combinatorial optimization over two random point sets · Zbl 1401.90180
[5] Barthe, F.; Cordero-Erausquin, D.; Maurey, B., Entropy of spherical marginals and related inequalities, J. Math. Pures Appl. (9), 86, 2, 89-99 (2006) · Zbl 1274.62059
[6] Ben Arous, G.; Zeitouni, O., Increasing propagation of chaos for mean field models, Ann. Inst. H. Poincaré Probab. Statist., 35, 1, 85-102 (1999) · Zbl 0928.60092
[7] Berry, A. C., The accuracy of the Gaussian approximation to the sum of independent variates, Trans. Amer. Math. Soc., 49, 122-136 (1941) · JFM 67.0461.01
[8] Bobkov, S. G.; Chistyakow, G.; Götze, F., Berry-Esseen bounds in the entropic central limit theorem · Zbl 1307.60011
[9] Bobkov, S. G.; Gentil, I.; Ledoux, M., Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl. (9), 80, 7, 669-696 (2001) · Zbl 1038.35020
[10] Boissard, E.; Le Gouic, T., On the mean speed of convergence of empirical and occupation measures in Wasserstein distance · Zbl 1294.60005
[11] Bolley, F.; Guillin, A.; Malrieu, F., Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, M2AN Math. Model. Numer. Anal., 44, 5, 867-884 (2010) · Zbl 1201.82029
[12] Bolley, F.; Guillin, A.; Villani, C., Quantitative concentration inequalities for empirical measures on non-compact spaces, Probab. Theory Related Fields, 137, 3-4, 541-593 (2007) · Zbl 1113.60093
[13] Caglioti, E.; Lions, P.-L.; Marchioro, C.; Pulvirenti, M., A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Comm. Math. Phys., 143, 3, 501-525 (1992) · Zbl 0745.76001
[14] Caglioti, E.; Lions, P.-L.; Marchioro, C.; Pulvirenti, M., A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. II, Comm. Math. Phys., 174, 2, 229-260 (1995) · Zbl 0840.76002
[15] Cantelli, F. P., Sulla determinazione empirica delle leggi di probabilita, Giorn. Ist. Ital. Attuari, 4, 421-424 (1933) · JFM 59.1166.05
[16] Carlen, E. A., Superadditivity of Fisher’s information and logarithmic Sobolev inequalities, J. Funct. Anal., 101, 1, 194-211 (1991) · Zbl 0732.60020
[17] Carlen, E. A.; Carvalho, M. C.; Le Roux, J.; Loss, M.; Villani, C., Entropy and chaos in the Kac model, Kinet. Relat. Models, 3, 1, 85-122 (2010) · Zbl 1186.76675
[18] Carlen, E. A.; Lieb, E. H.; Loss, M., A sharp analog of Young’s inequality on \(S^N\) and related entropy inequalities, J. Geom. Anal., 14, 3, 487-520 (2004) · Zbl 1056.43002
[19] Carrapatoso, K., Quantitative and qualitative Kac’s chaos on the Boltzmann’s sphere · Zbl 1341.60122
[20] Carrapatoso, K., Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules · Zbl 1332.82075
[21] Carrillo, J.; Toscani, G., Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Parma, 6, 75-198 (2007) · Zbl 1142.82018
[22] Choquet, G.; Meyer, P.-A., Existence et unicité des représentations intégrales dans les convexes compacts quelconques, Ann. Inst. Fourier (Grenoble), 13, 139-154 (1963) · Zbl 0122.34602
[23] Cordero-Erausquin, D., Some applications of mass transport to Gaussian-type inequalities, Arch. Ration. Mech. Anal., 161, 3, 257-269 (2002) · Zbl 0998.60080
[24] de Finetti, B., La prévision: ses lois logiques, ses sources subjectives, Ann. Inst. H. Poincaré, 7, 1, 1-68 (1937) · JFM 63.1070.02
[25] Diaconis, P.; Freedman, D., A dozen de Finetti-style results in search of a theory, Ann. Inst. H. Poincaré Probab. Statist., 23, 2 Suppl., 397-423 (1987) · Zbl 0619.60039
[26] Dobrić, V.; Yukich, J. E., Asymptotics for transportation cost in high dimensions, J. Theoret. Probab., 8, 1, 97-118 (1995) · Zbl 0811.60022
[27] Dobrušin, R. L., Vlasov equations, Funktsional. Anal. i Prilozhen., 13, 2, 48-58 (1979), 96
[28] Dudley, R. M., The speed of mean Glivenko-Cantelli convergence, Ann. Math. Statist., 40, 40-50 (1968) · Zbl 0184.41401
[29] Esseen, C.-G., On the Liapounoff limit of error in the theory of probability, Ark. Mat. Astr. Fys., 28A, 9 (1942), 19 pp · JFM 68.0277.02
[30] Feller, W., An Introduction to Probability Theory and Its Applications, vol. II (1971), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York · Zbl 0219.60003
[31] Fonseca, I.; Parry, G., Equilibrium configurations of defective crystals, Arch. Ration. Mech. Anal., 120, 3, 245-283 (1992) · Zbl 0785.73007
[33] Glivenko, V., Sulla determinazione empirica della legge di probabilita, Giorn. Ist. Ital. Attuari, 4, 92-99 (1933) · JFM 59.1166.04
[34] Goudon, T.; Junca, S.; Toscani, G., Fourier-based distances and Berry-Esseen like inequalities for smooth densities, Monatsh. Math., 135, 2, 115-136 (2002) · Zbl 0992.60027
[35] Graham, C.; Méléard, S., Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Ann. Probab., 25, 115-132 (1997) · Zbl 0873.60076
[36] Grünbaum, F. A., Propagation of chaos for the Boltzmann equation, Arch. Ration. Mech. Anal., 42, 323-345 (1971) · Zbl 0236.45011
[37] Guo, M. Z.; Papanicolaou, G. C.; Varadhan, S. R.S., Nonlinear diffusion limit for a system with nearest neighbor interactions, Comm. Math. Phys., 118, 1, 31-59 (1988) · Zbl 0652.60107
[38] Hauray, M.; Jabin, P.-E., \(N\)-particles approximation of the Vlasov equations with singular potential, Arch. Ration. Mech. Anal., 183, 3, 489-524 (2007) · Zbl 1107.76066
[40] Hewitt, E.; Savage, L. J., Symmetric measures on Cartesian products, Trans. Amer. Math. Soc., 80, 470-501 (1955) · Zbl 0066.29604
[41] Kac, M., Foundations of kinetic theory, (Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. III. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. III, Berkeley and Los Angeles, 1956 (1956), University of California Press), 171-197
[42] Kosygina, E., The behavior of the specific entropy in the hydrodynamic scaling limit, Ann. Probab., 29, 3, 1086-1110 (2001) · Zbl 1018.60096
[43] Lanford, O. E., Time evolution of large classical systems, (Dynamical Systems, Theory and Applications. Dynamical Systems, Theory and Applications, Recontres, Battelle Res. Inst., Seattle, WA, 1974. Dynamical Systems, Theory and Applications. Dynamical Systems, Theory and Applications, Recontres, Battelle Res. Inst., Seattle, WA, 1974, Lecture Notes in Phys., vol. 38 (1975), Springer: Springer Berlin), 1-111
[44] Lions, P.-L., Théorie des jeux de champ moyen et applications (mean field games), in: Cours du Collège de France, 2007-2009
[45] Lions, P.-L.; Toscani, G., A strengthened central limit theorem for smooth densities, J. Funct. Anal., 129, 1, 148-167 (1995) · Zbl 0822.60018
[46] Lott, J.; Villani, C., Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169, 3, 903-991 (2009) · Zbl 1178.53038
[47] Malrieu, F.; Logarithmic, Sobolev inequalities for some nonlinear PDE’s, Stochastic Process. Appl., 95, 1, 109-132 (2001) · Zbl 1059.60084
[48] McKean, H. P., Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas, Arch. Ration. Mech. Anal., 21, 343-367 (1966) · Zbl 1302.60049
[49] McKean, H. P., An exponential formula for solving Boltzmann’s equation for a Maxwellian gas, J. Combin. Theory, 2, 358-382 (1967) · Zbl 0152.46501
[50] McKean, H. P., The central limit theorem for Carleman’s equation, Israel J. Math., 21, 1, 54-92 (1975) · Zbl 0315.60013
[51] Mehler, F. G., Ueber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaschen Functionen höherer Ordnungn, Crelle’s J., 66, 161-176 (1866) · ERAM 066.1720cj
[52] Méléard, S., Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, (Probabilistic Models for Nonlinear Partial Differential Equations. Probabilistic Models for Nonlinear Partial Differential Equations, Montecatini Terme, 1995. Probabilistic Models for Nonlinear Partial Differential Equations. Probabilistic Models for Nonlinear Partial Differential Equations, Montecatini Terme, 1995, Lecture Notes in Math., vol. 1627 (1996), Springer: Springer Berlin), 42-95 · Zbl 0864.60077
[53] Messer, J.; Spohn, H., Statistical mechanics of the isothermal Lane-Emden equation, J. Stat. Phys., 29, 3, 561-578 (1982)
[54] Mischler, S., Le programme de Kac sur les limites de champ moyen, (Séminaire EDP-X (Décembre 2010))
[55] Mischler, S.; Mouhot, C., Kac’s program in kinetic theory, Invent. Math., 193, 1, 1-147 (2014), in press · Zbl 1274.82048
[58] Olla, S.; Varadhan, S. R.S.; Yau, H.-T., Hydrodynamical limit for a Hamiltonian system with weak noise, Comm. Math. Phys., 155, 3, 523-560 (1993) · Zbl 0781.60101
[59] Osada, H., Propagation of chaos for the two-dimensional Navier-Stokes equation, Proc. Japan Acad. Ser. A Math. Sci., 62, 1, 8-11 (1986) · Zbl 0679.76033
[60] Otto, F., The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26, 1-2, 101-174 (2001) · Zbl 0984.35089
[61] Otto, F.; Villani, C., Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173, 2, 361-400 (2000) · Zbl 0985.58019
[62] Rachev, S. T.; Rüschendorf, L., Mass transportation problems. vol. II, (Applications. Applications, Probab. Appl. (N. Y.) (1998), Springer-Verlag: Springer-Verlag New York) · Zbl 0990.60500
[63] Rio, E., Upper bounds for minimal distances in the central limit theorem, Ann. Inst. H. Poincaré Probab. Statist., 45, 3, 802-817 (2009) · Zbl 1175.60020
[64] Robinson, D. W.; Ruelle, D., Mean entropy of states in classical statistical mechanics, Comm. Math. Phys., 5, 288-300 (1967) · Zbl 0144.48205
[65] Sznitman, A.-S., Équations de type de Boltzmann, spatialement homogènes, Z. Wahrsch. Verw. Gebiete, 66, 4, 559-592 (1984) · Zbl 0553.60069
[66] Sznitman, A.-S., A propagation of chaos result for Burgers’ equation, (Hydrodynamic Behavior and Interacting Particle Systems. Hydrodynamic Behavior and Interacting Particle Systems, Minneapolis, MN, 1986. Hydrodynamic Behavior and Interacting Particle Systems. Hydrodynamic Behavior and Interacting Particle Systems, Minneapolis, MN, 1986, IMA Vol. Math. Appl., vol. 9 (1987), Springer: Springer New York), 181-188 · Zbl 0597.60055
[67] Sznitman, A.-S., Topics in propagation of chaos, (École d’Été de Probabilités de Saint-Flour XIX—1989. École d’Été de Probabilités de Saint-Flour XIX—1989, Lecture Notes in Math., vol. 1464 (1991), Springer: Springer Berlin), 165-251
[68] Talagrand, M., Matching random samples in many dimensions, Ann. Appl. Probab., 2, 4, 846-856 (1992) · Zbl 0761.60007
[69] Tanaka, H., Some probabilistic problems in the spatially homogeneous Boltzmann equation, (Theory and Application of Random Fields. Theory and Application of Random Fields, Bangalore, 1982. Theory and Application of Random Fields. Theory and Application of Random Fields, Bangalore, 1982, Lecture Notes in Control and Inform. Sci., vol. 49 (1983), Springer: Springer Berlin), 258-267
[70] Toscani, G., New a priori estimates for the spatially homogeneous Boltzmann equation, Contin. Mech. Thermodyn., 4, 2, 81-93 (1992) · Zbl 0760.76081
[71] Varadarajan, V. S., On the convergence of sample probability distributions, Sankhyā, 19, 23-26 (1958) · Zbl 0082.34201
[72] Villani, C., Fisher information estimates for Boltzmann’s collision operator, J. Math. Pures Appl., 77, 8, 821-837 (1998) · Zbl 0918.60093
[73] Villani, C., Decrease of the Fisher information for solutions of the spatially homogeneous Landau equation with Maxwellian molecules, Math. Models Methods Appl. Sci., 10, 2, 153-161 (2000) · Zbl 1010.82023
[74] Villani, C., Topics in Optimal Transportation, Grad. Stud. Math., vol. 58 (2003), American Mathematical Society · Zbl 1106.90001
[75] Villani, C., Optimal Transport, Old and New, Grundlehren Math. Wiss., vol. 338 (2009), Springer-Verlag: Springer-Verlag Berlin · Zbl 1156.53003
[76] Yau, H.-T., Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys., 22, 1, 63-80 (1991) · Zbl 0725.60120
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.