Geodesic
Fluctuations Pour Des Équations De Boltzmann Scalaires
Canadian journal of mathematics, Tome 43 (1991) no. 5, pp. 975-984

Voir la notice de l'article provenant de la source Cambridge University Press

Kac [22] a montré, pour sa caricature d'un gaz de Boltzmann, que la solution de l'équation de Boltzmann généralisée correspondante peut être obtenue comme limite d'une suite de lois empiriques induites par des processus markoviens . Pour un entier n donné, le processus (Xn (t))t≥0 décrit le comportement d'un gaz à n molécules où des collisions binaires ont lieu à des instants imprévisibles. Ce processus est gouverné par un générateur Gn qui est défini à l'aide de l'opérateur de collision apparaissant dans la version de Kac de l'équation de Boltzmann. 
Ferland, René. Fluctuations Pour Des Équations De Boltzmann Scalaires. Canadian journal of mathematics, Tome 43 (1991) no. 5, pp. 975-984. doi: 10.4153/CJM-1991-054-5
@article{10_4153_CJM_1991_054_5,
     author = {Ferland, Ren\'e},
     title = {Fluctuations {Pour} {Des} {\'Equations} {De} {Boltzmann} {Scalaires}},
     journal = {Canadian journal of mathematics},
     pages = {975--984},
     year = {1991},
     volume = {43},
     number = {5},
     doi = {10.4153/CJM-1991-054-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-054-5/}
}
TY  - JOUR
AU  - Ferland, René
TI  - Fluctuations Pour Des Équations De Boltzmann Scalaires
JO  - Canadian journal of mathematics
PY  - 1991
SP  - 975
EP  - 984
VL  - 43
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-054-5/
DO  - 10.4153/CJM-1991-054-5
ID  - 10_4153_CJM_1991_054_5
ER  - 
%0 Journal Article
%A Ferland, René
%T Fluctuations Pour Des Équations De Boltzmann Scalaires
%J Canadian journal of mathematics
%D 1991
%P 975-984
%V 43
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-054-5/
%R 10.4153/CJM-1991-054-5
%F 10_4153_CJM_1991_054_5

[1] 1. Barnsley, M.F. and Turchetti, G., A Study ofBoltzmann energy equations, Ann. Phys. 159 (1985), 1–61. Google Scholar

[2] 2. Billingsley, P., Convergence of probability measures. Wiley, New York, 1968. Google Scholar

[3] 3. Bobylev, A.V., Exact solutions of the Boltzmann equation, Soviet Phys. Dokl. 20 (1976), 822–824. Google Scholar

[4] 4. Bobylev, A.V., A class of invariant solutions of the Boltzmann equation, Soviet Phys. Dokl. 21 (1976), 632–634. Google Scholar

[5] 5. Braun, W. and Hepp, K., The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles, Commun. Math. Phys. 56 (1977), 101–113. Google Scholar

[6] 6. Chapman, S. and Cowling, T.G., The Mathematical theory of non-uniform gases. Cambridge University Press, London, 1970. Google Scholar

[7] 7. Duderstadt, J.J. and Martin, W.R., Transport theory. Wiley, New York, 1979. Google Scholar

[8] 8. Ernst, M.H., Nonlinear model-Boltzmann equations and exact solutions, Phys. Rep. 78 (1981), 1–171. Google Scholar

[9] 9. Ernst, M.H. and Hendriks, E.M., An exactly solvable non-linear Boltzmann equation, Phys. Lett. A 70 (1979), 183–185. Google Scholar

[10] 10. Ethier, S.N. and Kurtz, T.G., Markov processes. Characterization and convergence. Wiley, New York, 1986. Google Scholar

[11] 11. Ferland, R. and Giroux, G., Cutoff-type Boltzmann equations: Convergence of the solution, Adv. in Appl. Math. 8 (1987), 98–107. Google Scholar

[12] 12. Ferland, R., Équations de Boltzmann scalaires : convergence de la solution, fluctuationset propagation du chaos trajectorielle. Thèse de doctorat, Université de Sherbrooke, 1990. Google Scholar

[13] 13. Fernique, X., Fonctions aléatoires à valeurs dans les espaces lusiniens, preprint. Google Scholar

[14] 14. Fernique, X., Vecteurs aléatoires lusiniens et fonctions aléatoires lusiniennes. Rapport de recherche no 67, Département de mathématiques et d'informatique, Université de Sherbrooke, 1989. Google Scholar

[15] 15. Futcher, F.J., Hoare, M.R., Hendriks, E.M. and Ernst, M.H., Soluble Boltzmann equations for internal state and Maxwell models, Phys. A 101 (1980), 185–204. Google Scholar

[16] 16. Futcher, F.J. and Hoare, M.R., The p-q model Boltzmann equation, Phys. A 122 (1983), 516–546. Google Scholar

[17] 17. Gartner, J., On the McKean-Vlasov limit for interacting diffusions, Math. Nachr. 137 (1988), 197–248. Google Scholar

[18] 18. Holley, R.A. and Stroock, D.V., Generalized Ornstein-Uhlenbeckprocesses and infinite particle branching Brownian motion, Publ. Res. Inst. Math. Sci. 14 (1978), 741–788. Google Scholar

[19] 19. Hoare, M.H., Quadratic transport and soluble Boltzmann equation, Adv. Chem. Phys. 56 (1984), 1–140. Google Scholar

[20] 20. Huxley, L.G.H. and Crompton, R.W., The Diffusion and drift of electrons in gases. Wiley, New York, 1974. Google Scholar

[21] 21. Jakubowski, A., On the Skorokhod topology, Ann. Inst. Henri Poincaré 22 (1986), 263–285. Google Scholar

[22] 22. Kac, M., Foundations of kinetic theory, Proc. Third Berkeley Symp. Math. Statist. Prob. (J. Neyman, ed.) 3 (1956), 171–197. Google Scholar

[23] 23. Kac, M., Some probabilistic aspects of the Boltzmann-equation, Acta Phys. Austriaca Suppl. 10 (1973), 379–400. Google Scholar

[24] 24. Kesten, H., Quadratic transformations: a model for population growth I, Adv. in Appl. Probab. 2 (1970), 1–82. Google Scholar

[25] 25. Kesten, H., Quadratic transformations: a model for population growth II, Adv. in Appl. Probab. 2 (1970), 179–228. Google Scholar

[26] 26. Krook, M. and Wu, T.T., Formation ofMaxwellian tails, Phys. Rev. Lett. 36 (1976), 1107–1109. Google Scholar

[27] 27. Krook, M. and Wu, T.T., Exact solutions of the Boltzmann equation, Phys. Fluids 20 (1977), 1589–159. Google Scholar

[28] 28. McKean, H.P., Fluctuations in the kinetic theory of gases, Comm. Pure Appl. Math. 28 (1975), 435–455. Google Scholar

[29] 29. Moran, P.A.P., The Statistical processes of evolutionary theory. Clarendon Press, Oxford, 1962. Google Scholar

[30] 30. Nishimura, S., Random collision processes and their limiting distribution using the discrimination information, J. Appl. Probab. 11 (1974), 266–280. Google Scholar

[31] 31. Nishimura, S., Random collision processes with transition probabilities belonging to the same type of distribution, J. Appl. Probab. 11 (1974), 703–714. Google Scholar

[32] 32. Rankin, C.C. and Light, J.C., Relaxation of a gas of harmonic oscillators, J. Chem. Phys. 46 (1967), 1305-1316. Google Scholar

[33] 33. Shuler, K.E., Relaxation of an isolated ensemble of harmonic oscillators, J. Chem. Phys. 32 (1960), 1692- 1697. Google Scholar

[34] 34. Tanaka, H., Propagation of chaos for certain purely discontinuous Markov processes with interaction, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 17 (1970), 253–272. Google Scholar

[35] 35. Tanaka, H., Fluctuation theory for Kacs one-dimensional model ofMaxwellian molecules, Sankhyā Ser. A 44 (1982), 23–26. Google Scholar

[36] 36. Tanaka, H., Some probabilistic problems in the spatially homogeneous Boltzmann equation. Proc. of IFIP-ISI Conf. Theo. Appl. of Random Fields, Bangalore, 1982. Google Scholar

[37] 37. Tjon, J. and Wu, T.T., Numerical aspects of the approach to a Maxwellian distribution. Phys. Rev. A 19 (1979), 883–888. Google Scholar

[38] 38. Uchiyama, K., Fluctuations ofMarkovian systems in Kacs caricature of a Maxwellian gas, J. Math. Soc. Japan 35 (1983), 477–199. Google Scholar

[39] 39. Uchiyama, K., A fluctuation problem associated with the Boltzmann equation for a gas of molecules with a cutoff potential, Japan. J. Math. 9 (1983), 27–53. Google Scholar

[40] 40. Uchiyama, K., Fluctuations in population dynamics. Lecture Notes in Biomath. 70, Springer-Verlag, New York, 1986, 222–229. Google Scholar

[41] 41. Wu, T.T., Kinetic equations of gases and plasmas. Addison-Wesley, Reading, 1966. Google Scholar

Cité par Sources :