Abstract
The properties of the nonlinear collision integral in the Boltzmann equation are studied. Expansions in spherical Hermitean polynomials are used. It was shown [1] that the nonlinear matrix elements of the collision operator are related to each other by simple expressions, which are valid for arbitrary cross sections of particle interaction. The structure of the collision operator and the properties of the matrix elements are studied for the case when the interaction potential is spherically symmetric. In this case, the linear Boltzmann operator satisfies the Hecke theorem. The generalized Hecke theorem, from which it follows that many nonlinear matrix elements vanish, is proved with recurrence relations derived. It is shown that the generalized Hecke theorem is a consequence of the ordinary Hecke theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Ya. Énder and I. A. Énder, Zh. Tekh. Fiz. 72(5), 1 (2002) [Tech. Phys. 47, 513 (2002)].
A. Ya. Énder and I. A. Énder, Zh. Tekh. Fiz. 69(6), 22 (1999) [Tech. Phys. 44, 628 (1999)].
A. Yu. Ender and I. A. Ender, Phys. Fluids 11, 2720 (1999).
E. Hecke, Math. Ann. 78, 398 (1917).
E. Hecke, Math. Z. 12, 274 (1922).
K. Kumar, Ann. Phys. (N.Y.) 37, 113 (1966).
S. Chapman and T. G. Cowling, Mathematical Theory of Non-Uniform Gases, 2nd ed. (Cambridge Univ. Press, Cambridge, 1952; Inostrannaya Literatura, Moscow, 1960).
D. Burnett, Proc. London Math. Soc. 39, 385 (1935).
D. Burnett, Proc. London Math. Soc. 40 (1935).
H. Grad, Commun. Pure Appl. Math. 2, 311 (1949).
L. Sirovich, Phys. Fluids 6(1), 10 (1963).
I. A. Énder and A. Ya. Énder, Dokl. Akad. Nauk SSSR 193(1), 61 (1970) [Sov. Phys. Dokl. 15, 633 (1971)]; Phys. Fluids 6 (1), 10 (1963).
A. Ya. Énder, Zh. Tekh. Fiz. 62(1), 20 (1992) [Sov. Phys. Tech. Phys. 37, 9 (1992)].
D. Hilbert, Math. Ann. 72, 562 (1912).
I. Talmi, Helv. Phys. Acta 25, 185 (1952).
Yu. F. Smirnov, Nucl. Phys. 27, 177 (1961).
V. V. Vedenyapin, Preprint No. 38, IPM AN SSSR (Inst. of Applied Mathematics, USSR Academy of Sciences, 1981).
D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (Nauka, Leningrad, 1975; World Sci., Singapore, 1988).
E. M. Hendrics and T. M. Nieuwenhuizen, J. Stat. Phys. 29(3), 591 (1982).
V. V. Vedenyapin, Dokl. Akad. Nauk SSSR 256, 338 (1981) [Sov. Phys. Dokl. 26, 26 (1981)].
A. Ya. Énder and I. A. Énder, Aerodynamics of Rarefied Gases (Leningrad, 1983), pp. 197–215.
A. Ya. Énder, I. A. Énder, and M. B. Lyutenko, Preprint No. 1748, FTI RAN (Physicotechnical Inst., Russian Academy of Sciences, St. Petersburg, 2000).
J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport in Gases (North-Holland, Amsterdam, 1972; Mir, Moscow, 1976).
Author information
Authors and Affiliations
Additional information
__________
Translated from Zhurnal Tekhnichesko\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l}\) Fiziki, Vol. 73, No. 2, 2003, pp. 6–12.
Original Russian Text Copyright © 2003 by A. Énder, I. Énder.
Rights and permissions
About this article
Cite this article
Énder, A.Y., Énder, I.A. Generalization of the Hecke theorem for the nonlinear Boltzmann collision integral in the axisymmetric case. Tech. Phys. 48, 138–145 (2003). https://doi.org/10.1134/1.1553552
Received:
Issue Date:
DOI: https://doi.org/10.1134/1.1553552