From Newton to Boltzmann: Hard Spheres and Short-range Potentials

We provide a rigorous derivation of the Boltzmann equation as the mesoscopic limit of systems of hard spheres, or Newtonian particles interacting via a short-range potential, as the number of particles $N$ goes to infinity and the characteristic length of interaction $\e$ simultaneously goes to $0,$ in the Boltzmann-Grad scaling $N \e^{d-1} \equiv 1.$ The time of validity of the convergence is a fraction of the average time of first collision, due to a limitation of the time on which one can prove uniform estimates for the BBGKY and Boltzmann hierarchies. Our proof relies on the fundamental ideas of Lanford, and the important contributions of King, Cercignani, Illner and Pulvirenti, and Cercignani, Gerasimenko and Petrina. The main novelty here is the detailed study of pathological trajectories involving recollisions, which proves the term-by-term convergence for the correlation series expansion.

[1]  M. Pulvirenti,et al.  THE LINEAR BOLTZMANN EQUATION FOR LONG-RANGE FORCES: A DERIVATION FROM PARTICLE SYSTEMS , 1999 .

[2]  Julius,et al.  Works , 1867, Landscape Journal.

[3]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[4]  Max Born,et al.  A general kinetic theory of liquids , 1949 .

[5]  H. Grad.  Principles of the Kinetic Theory of Gases , 1958 .

[6]  M. Pulvirenti,et al.  Chapter 1 – From Particles to Fluids , 2005 .

[7]  P. Lions,et al.  On the Cauchy problem for Boltzmann equations: global existence and weak stability , 1989 .

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

[9]  K. Uchiyama.  Derivation of the Boltzmann equation from particle dynamics , 1988 .

[10]  C. Cercignani.  On the Boltzmann equation for rigid spheres , 1972 .

[11]  D. Haar,et al.  Statistical Physics , 1971, Nature.

[12]  W. Braun,et al.  The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles , 1977 .

[13]  S. Ukai,et al.  On the existence of global solutions of mixed problem for non-linear Boltzmann equation , 1974 .

[14]  N. N. Bogolyubov,et al.  Problems of a Dynamical Theory in Statistical Physics , 1959 .

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

[16]  Time evolution for infinitely many hard spheres , 1976 .

[17]  M. Pulvirenti,et al.  On the validity of the Boltzmann equation for short range potentials , 2013, 1301.2514.

[18]  C. Cercignani,et al.  Many-Particle Dynamics And Kinetic Equations , 1997 .

[19]  L. Desvillettes,et al.  A Rigorous Derivation of a Linear Kinetic Equation of Fokker–Planck Type in the Limit of Grazing Collisions , 2001 .

[20]  Louis Nirenberg,et al.  An abstract form of the nonlinear Cauchy-Kowalewski theorem , 1972 .

[21]  H. Grad.  On the kinetic theory of rarefied gases , 1949 .

[22]  S. Simonella.  Evolution of Correlation Functions in the Hard Sphere Dynamics , 2012, 1205.2789.

[23]  François Golse,et al.  The mean-field limit for the dynamics of large particle systems , 2003 .

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

[25]  Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum , 1986 .

[26]  D. Hilbert.  Sur les problèmes futurs des mathématiques : les 23 problèmes , 1990 .

[27]  Pierre-Louis Lions,et al.  Regularity of the moments of the solution of a Transport Equation , 1988 .

[28]  L. Saint-Raymond,et al.  The Brownian motion as the limit of a deterministic system of hard-spheres , 2013, 1305.3397.

[29]  Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum , 1986 .

[30]  M. Pulvirenti,et al.  Derivation of the Fick’s Law for the Lorentz Model in a Low Density Regime , 2014, 1404.4186.

[31]  J. Yvon,et al.  La théorie statistique des fluides et l'équation d'état , 1935 .

[32]  H. Spohn.  Large Scale Dynamics of Interacting Particles , 1991 .

[33]  Hans J. Herrmann,et al.  Particles in fluids , 2007, Comput. Phys. Commun..

[34]  Béla Bollobás,et al.  Random Graphs , 1985 .

[35]  Ludwig Boltzmann.  Leçons sur la théorie des gaz , 1992 .

[36]  H. Georgii.  The equivalence of ensembles for classical systems of particles , 1995 .

[37]  H. Spohn.  Boltzmann hierarchy and boltzmann equation , 1984 .

[38]  Radjesvarane Alexandre,et al.  On the Boltzmann equation for long‐range interactions , 2002 .

[39]  S. Ukai.  The Boltzmann-Grad limit and Cauchy-Kovalevskaya theorem , 2001 .

[40]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[41]  Takaaki Nishida,et al.  A note on a theorem of Nirenberg , 1977 .

[42]  C. Villani.  Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory , 2002 .

[43]  R. Illner,et al.  Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum: Erratum and improved result , 1989 .