Summary
We study one dimensional particle systems in which particles travel as independent random walks and collide stochastically. The collision rates are chosen so that each particle experiences finitely many collisions per unit time. We establish the kinetic limit and derive the discrete Boltzmann equation for the macroscopic particle density.
References
[B] Bony, J.-M.: Solutions globales bornées pour les modèles discrets de l'equation de Boltzmann en dimension 1 d'espace. Actes Journées E.D.P. St. Jean de Monts, no. XVI (1987)
[CDPP] Caprino, S., DeMasi, A., Presutti, E., Pulvirenti, M.: A derivation of the Broadwell equation. Comm. Math. Phys.135, 443–465 (1991)
[CP] Caprino, S., Pulvirenti, M.: A cluster expansion approach to a one-dimensional Boltzmann equation: a validity result. Preprint
[DP] DeMasi, A., Presutti, E.: Mathematical methods for hydrodynamical limits (Lect. Notes Math., vol. 1501) Berlin Heidelberg New York: Springer 1991
[EK] Ethier, S.N., Kurtz, G. T.: Markov processes, characterization and convergence. New York: Wiley 1980
[IPI] Illner, R., Platkowski, T.: Discrete velocity models of the Boltzmann equation: a survey on the mathematical aspects of the theory. SIAM Rev.30, 231–255 (1988)
[IPul] Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum. Comm. Math. Phys.105, 189–203 (1986)
[IPu2] Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for a two and three-dimensional rare gas in vacuum: erratum and improved results. Comm. Math. Phys.121, 143–146 (1989)
[K] King, F.: BBGKY hierarchy for positive potentials. Ph.D. Thesis, Department of Mathematies, University of California at Berkeley (1975)
[L] Lanford, O.E.: Time evolution of large classical systems. In: Moser, M.J. (Lect. Notes Phys. vol. 38, pp. 1–111) Berlin: Springer 1975
[P] Pulvirenti, M.: Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum. Comm. Math. Phys.113, 79–85 (1987)
[R] Rezakhanlou, F.: Propagation of chaos for particle systems associated with discrete Boltzmann equation. Preprint
[RT] Rezakhanlou, F., Tarver, J.: in preparation
[Spi] Spitzer, F.: Principles of random walks, 2nd edition, Berlin Heidelberg New York: Springer 1976
[Spo] Spohn, H.: Large scale dynamics of interacting particles. Berlin Heidelberg New York: Springer 1991
[T] Tartar, L.: Some existence theorems for semilinear hyperbolic systems in one space variables. University of Wisconsin, MRC Technical Summary Report (1980)
[H] Yau, H.-T.: Metastability of Ginzburg-Landau model with a conservation law. J. Stat. Phys.74, 705–742
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Rezakhanlon, F. Kinetic limits for a class of interacting particle systems. Probab. Th. Rel. Fields 104, 97–146 (1996). https://doi.org/10.1007/BF01303805
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DOI: https://doi.org/10.1007/BF01303805
Mathematics Subject Classification
- 60K35
- 76P05
- 82C40