On Kac's chaos and related problems

This paper is devoted to establish quantitative and qualitative estimates related to the notion of chaos as firstly formulated by M. Kac \cite{Kac1956} in his study of mean-field limit for systems of $N$ undistinguishable particles as $N\to\infty$. First, we quantitatively liken three usual measures of {\it Kac's chaos}, some involving the all $N$ variables, other involving a finite fixed number of variables. The cornerstone of the proof is a new representation of the Monge-Kantorovich-Wasserstein (MKW) distance for symmetric $N$-particle probability measures in terms of the distance between the law of the associated empirical measures on the one hand, and a new estimate on some MKW distance on probability measures spaces endowed with a suitable Hilbert norm taking advantage of the associated good algebraic structure. Next, we define the notion of {\it entropy chaos } and {\it Fisher information chaos } in a similar way as defined by Carlen et al \cite{CCLLV}. We show that {\it Fisher information chaos } is stronger than {\it entropy chaos}, which in turn is stronger than {\it Kac's chaos}. More importantly, with the help of the HWI inequality of Otto-Villani, we establish a quantitative estimate between these quantities, which in particular asserts that {\it Kac's chaos} plus {\it Fisher information bound } implies {\it entropy chaos}. We then extend the above quantitative and qualitative results about chaos in the framework of probability measures with support on the {\it Kac's spheres}, revisiting \cite{CCLLV} and giving a possible answer to \cite[Open problem 11]{CCLLV}. Additionally to the above mentioned tool, we use and prove an optimal rate local CLT in $L^\infty$ norm for distributions with finite 6-th moment and finite $L^p$ norm, for some $p>1$. Last, we investigate how our techniques can be used without assuming chaos, in the context of probability measures mixtures introduced by De Finetti, Hewitt and Savage. In particular, we define the (level 3) Fisher information for mixtures and prove that it is l.s.c. and affine, as that was done in \cite{RR} for the level 3 Boltzmann's entropy.

[1]  E. Carlen.  Superadditivity of Fisher's information and logarithmic Sobolev inequalities , 1991 .

[2]  J. Doob.  Stochastic processes , 1953 .

[3]  P. Meyer,et al.  Existence et unicité des représentations intégrales dans les convexes compacts quelconques , 1963 .

[4]  Jo Graham,et al.  Old and new , 2000 .

[5]  A. Guillin,et al.  Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation , 2009, 0906.1417.

[6]  Elena Kosygina.  THE BEHAVIOR OF THE SPECIFIC ENTROPY IN THE HYDRODYNAMIC SCALING LIMIT , 2001 .

[7]  C. Bordenave,et al.  Combinatorial Optimization Over Two Random Point Sets , 2011, 1103.2734.

[8]  C. Villani.  Optimal Transport: Old and New , 2008 .

[9]  C. Villani,et al.  Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality , 2000 .

[10]  F. Otto.  THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION , 2001 .

[11]  Gennadiy P. Chistyakov,et al.  Berry–Esseen bounds in the entropic central limit theorem , 2011, 1105.4119.

[12]  G. Toscani,et al.  Contractive Probability Metrics and Asymptotic Behavior of Dissipative Kinetic Equations , 2007 .

[13]  F. Barthe,et al.  Entropy of spherical marginals and related inequalities , 2006 .

[14]  F. Grünbaum,et al.  Propagation of chaos for the Boltzmann equation , 1971 .

[15]  S. Varadhan,et al.  Hydrodynamical limit for a Hamiltonian system with weak noise , 1993 .

[16]  H. McKean.  Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas , 1966 .

[17]  C. Villani,et al.  Entropy and chaos in the Kac model , 2008, 0808.3192.

[18]  C. Villani.  Topics in Optimal Transportation , 2003 .

[19]  O. Lanford.  Time evolution of large classical systems , 1975 .

[20]  Pierre-Emmanuel Jabin,et al.  N-particles Approximation of the Vlasov Equations with Singular Potential , 2003, math/0310039.

[21]  Henry P. McKean,et al.  The central limit theorem for Carleman’s equation , 1975 .

[22]  Thierry Goudon,et al.  Fourier-Based Distances and Berry-Esseen Like Inequalities for Smooth Densities , 2002 .

[23]  P. Jabin,et al.  Particles approximations of Vlasov equations with singular forces : Part. 2 , 2010 .

[24]  Michel Talagrand,et al.  Matching Random Samples in Many Dimensions , 1992 .

[25]  J. Yukich,et al.  Asymptotics for transportation cost in high dimensions , 1995 .

[26]  R. Dudley.  The Speed of Mean Glivenko-Cantelli Convergence , 1969 .

[27]  L. J. Savage,et al.  Symmetric measures on Cartesian products , 1955 .

[28]  C. Villani,et al.  Quantitative Concentration Inequalities for Empirical Measures on Non-compact Spaces , 2005, math/0503123.

[29]  F. G. Mehler.  Ueber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaceschen Functionen höherer Ordnung. , 1866 .

[30]  DECREASE OF THE FISHER INFORMATION FOR SOLUTIONS OF THE SPATIALLY HOMOGENEOUS LANDAU EQUATION WITH MAXWELLIAN MOLECULES , 2000 .

[31]  C. Villani,et al.  Ricci curvature for metric-measure spaces via optimal transport , 2004, math/0412127.

[32]  Gennadiy P. Chistyakov,et al.  Rate of convergence and Edgeworth-type expansion in the entropic central limit theorem , 2011, 1104.3994.

[33]  W. Feller,et al.  An Introduction to Probability Theory and Its Applications, Vol. II , 1972, The Mathematical Gazette.

[34]  Horng-Tzer Yau,et al.  Relative entropy and hydrodynamics of Ginzburg-Landau models , 1991 .

[35]  H. McKean.  An exponential formula for solving Boltzmann's equation for a Maxwellian gas* , 1967 .

[36]  G. Toscani.  New a priori estimates for the spatially homogeneous Boltzmann equation , 1992 .

[37]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[38]  I. Fonseca,et al.  Equilibrium configurations of defective crystals , 1992 .

[39]  D. Freedman,et al.  A dozen de Finetti-style results in search of a theory , 1987 .

[40]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[41]  S. Bobkov,et al.  Hypercontractivity of Hamilton-Jacobi equations , 2001 .

[42]  Derek W. Robinson,et al.  Mean entropy of states in classical statistical mechanics , 1967 .

[43]  C. Chou.  The Vlasov equations , 1965 .

[44]  Thibaut Le Gouic,et al.  On the mean speed of convergence of empirical and occupation measures in Wasserstein distance , 2011, 1105.5263.

[45]  George Papanicolaou,et al.  Nonlinear diffusion limit for a system with nearest neighbor interactions , 1988 .

[46]  École d'été de probabilités de Saint-Flour,et al.  Ecole d'été de probabilités de Saint-Flour XIX, 1989 , 1991 .

[47]  A. C. Berry.  The accuracy of the Gaussian approximation to the sum of independent variates , 1941 .

[48]  Cédric Villani,et al.  Fisher information estimates for Boltzmann's collision operator , 1998 .

[49]  K. Carrapatoso.  Propagation of chaos for the spatially homogeneous Landau equation for maxwellian molecules , 2012, 1212.3724.

[50]  S. Méléard.  Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models , 1996 .

[51]  Emanuele Caglioti,et al.  A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description , 1992 .

[52]  Ofer Zeitouni,et al.  Increasing propagation of chaos for mean field models , 1999 .

[53]  Pierre-Louis Lions,et al.  A Strengthened Central Limit Theorem for Smooth Densities , 1995 .

[54]  Florent Malrieu,et al.  Logarithmic Sobolev Inequalities for Some Nonlinear Pde's , 2001 .

[55]  Dario Cordero-Erausquin,et al.  Some Applications of Mass Transport to Gaussian-Type Inequalities , 2002 .

[56]  A. Sznitman.  Topics in propagation of chaos , 1991 .

[57]  S. Rachev,et al.  Mass transportation problems , 1998 .

[58]  The homogeneous Boltzmann hierarchy and statistical solutions to the homogeneous Boltzmann equation , 1991 .

[59]  A. Sznitman.  Équations de type de Boltzmann, spatialement homogènes , 1984 .

[60]  Hiroshi Tanaka,et al.  Some probabilistic problems in the spatially homogeneous Boltzmann equation , 1983 .

[61]  Nicolas Fournier,et al.  Propagation of chaos for the 2D viscous vortex model , 2012, 1212.1437.

[62]  Herbert Spohn,et al.  Statistical mechanics of the isothermal lane-emden equation , 1982 .

[63]  S. A.,et al.  A Propagation of Chaos Result for Burgers ' Equation , 2022 .

[64]  K. Carrapatoso.  Quantitative and qualitative Kac's chaos on the Boltzmann's sphere , 2012, 1205.1241.

[65]  János Komlós,et al.  On optimal matchings , 1984, Comb..

[66]  E. Lieb,et al.  A sharp analog of Young’s inequality on SN and related entropy inequalities , 2004, math/0408030.

[67]  Hirofumi Osada,et al.  Propagation of chaos for the two dimensional Navier-Stokes equation , 1986 .

[68]  B. D. Finetti.  La prévision : ses lois logiques, ses sources subjectives , 1937 .

[69]  C. Graham,et al.  Stochastic particle approximations for generalized Boltzmann models and convergence estimates , 1997 .

[70]  M. Kac.  Foundations of Kinetic Theory , 1956 .

[71]  Assaf Naor,et al.  On the rate of convergence in the entropic central limit theorem , 2004 .

[72]  E. Rio.  Upper bounds for minimal distances in the central limit theorem , 2009 .