Abstract
In this expository article, we discuss four conjectures concerning the kinetic behavior of the hard sphere models. We then formulate four stochastic variations of the hard sphere model. The known results for these stochastic models are reviewed and some proofs are sketched.
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References
H. van Beijeren, O. E. Lanford, J. L. Lebowitz and H. Spohn,Equilibrium time correlation functions in the low-density limit. J. Statist. Phys. 22 (1980), 237-257.
J-M. Bony, Solutions globales bornées pour les modèles discrets de l’équation de Boltzmann en dimension 1 d’espace. In Journées “Equations aux Derivées Partielles” Saint-Jean-de-Monts, 1987.
S. Caprino, A. De Masi, E. Presutti, M. Pulvirenti, A derivation of the Broadwell equation. Comm. Math. Phys. 135 (1991), 443-465.
S. Caprino and M. Pulvirenti, A cluster expansion approach to a one-dimensional Boltzmann equation: a validity result. Comm. Math. Phys. 166 (1995), 603-631.
C. Cercignani, R. Illner and M. Pulvirenti, “The Mathematical Theory of Dilute Gases”, Springer-Verlag, 1994.
R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Annals of Math. 130 (1989), 321-366.
R. J. DiPerna and P. L. Lions, Global solutions of Boltzmann equation and the entropy inequality, Arch. Rational Mech. Anal. 114 (1991), 47-55.
F. Golse, P. L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Func. Anal. 76 (1988), 110-125.
R. Illner and M. Pulvirenti,Global validity of the Boltzmann equation for a two and three-dimensional rare gas in vacuum, Comm. Math. Phys. 105 (1986), 189-203. Erratum and improved results, Comm. Math. Phys. 121 (1989), 143-146.
F. King,BBGKY hierarchy for positive potentials, Ph.D. Thesis, Dept. of Mathematics, University of California at Berkeley, (1975).
O. E. Lanford (III),Time evolution of large classical systems, in “Lecture Notes in Physics”, editor J. Moser, vol. 38, Springer-Verlag, Berlin, 1975.
P. L. Lions, Compactness in Boltzmann’s equation via Fourier integral operators and applications I, II, III, J. Math. Kyoto Univ. 34 (1994), 391-427, 429-461, 539-584.
M. Pulvirenti, Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum, Comm. Math. Phys. 113 (1987), 79-85.
P. Resibois and M. De Leener, “Classical Kinetic Theory of Fluids”. Wiley, New York 1977.
F. Rezakhanlou, Kinetic limits for a class of interacting particle systems, Probab. Theory Related Fields,104 (1996), 97-146.
F. Rezakhanlou, Propagation of chaos for particle systems associated with discrete Boltzmann equation, Stochastic Processes and their Applications, 64 (1996), 55-72.
F. Rezakhanlou, Equilibrium fluctuations for the discrete Boltzmann equation, Duke Journal of Mathematics, 93 (1998), 257-288.
F. Rezakhanlou, Large deviations from a kinetic limit, Annals of Probability, 26 (1998), 1259-1340.
F. Rezakhanlou, A stochastic model associated with Enskog equation and its kinetic limit, Comm. Math. Phys. 232 (2003), 327-375.
F. Rezakhanlou, Boltzmann-Grad limits for stochastic hard sphere model, Comm. Math. Phys. 248 (2004), 553-637.
F. Rezakhanlou and J. L. Tarver (III),Boltzmann-Grad limit for a particle system in continuum, Ann. Inst. Henri Poincaré ,33 (1997), 753-796.
H. Spohn, “Large Scale Dynamics of Interacting Particles”, Springer-Verlag, 1991.
H. Spohn, Fluctuations around the Boltzmann equation, J. Statist. Phys. 26 (1981), 285-305.
L. Tartar, Existence globale pour un système hyperbolique semilinéaire de la théorie cinétique des gaz, Séminaire Goulaouic-Schwartz (1975/1976), 1976.
L. Tartar, Some existence theorems for semilinear hyperbolic systems in one space variable, Technical report, University of Wisconsin-Madison, 1980.
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Rezakhanlou, F. (2008). Kinetic Limits for Interacting Particle Systems. In: Golse, F., Olla, S. (eds) Entropy Methods for the Boltzmann Equation. Lecture Notes in Mathematics, vol 1916. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73705-6_2
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DOI: https://doi.org/10.1007/978-3-540-73705-6_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73704-9
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