Abstract
Although the Chapman-Enskog treatment of the Boltzmann equation is one of the first examples of the elimination of fast variables, it is not usually presented consistently from that point of view. Here it is developed systematically as a special case of the general method for eliminating fast variables from nonlinear equations. As a result certain ambiguities can be remedied. First, it is inconsistent with the separation of time scales to extend the phenomenological description of the gas by including some of the higher moments of the distribution function (such as the heat flow). In the application to the relativistic Boltzmann equation, the dilemma concerning the choice of the lowest order approximation is resolved. In the final section it is demonstrated that unsystematic elimination of fast variables leads to secular terms.
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References
S. Chapman,Proc. R. Soc. Lond. A 93:1 (1916–1917); D. Enskog, Thesis, Uppsala (1917) [Both reprinted in S. G. Brush,Kinetic Theory, Vol. 3 (Pergamon, Oxford, 1972)].
S. Chapman and T. G. Cowling,The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, 1939).
H. Grad, inHandbuch der Physik, Vol. 12, S. Flügge, ed. (Springer, Berlin, 1958).
P. Résibois and M. de Leener,Classical Kinetic Theory of Fluids (Wiley-Interscience, New York, 1977).
G. E. Uhlenbeck and G. W. Ford,Lectures in Statistical Mechanics (American Mathematical Society, Providence, Rhode Island, 1963); C. Cercignani,Mathematical Methods in Kinetic Theory (Plenum Press, New York, 1969), Chapters 5 and 6; C. Cercignani,Theory and Application of the Boltzmann Equation (Scottish Academic Press, Edinburgh, 1975).
C. F. Curtiss and J. O. Hirschfelder,J. Chem. Phys. 17:550 (1949); J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird,Molecular Theory of Gases and Liquids (Wiley, New York, 1954).
U. M. Titulaer,Physica 91A:321 (1978); U. Geigernnüller, U. M. Titulaer, and B. U. Felderhof,Physica 119A:41, 53 (1983).
C. W. Gardiner,Handbook of Stochastic Methods (Springer, Berlin, 1983), p. 218.
H. Haken,Handbuch des Physik, Vol. 25/2c, S. Flügge, ed. (Springer, Berlin, 1970); H. Haken,Synergetics, 2nd ed. (Springer, Berlin, 1978), p. 194.
N. G. van Kampen,Phys. Rep. 124:69 (1985).
U. M. Titulaer,Physica 100A:234 (1980).
R. E. Nettleton,J. Chem. Phys. 40:112 (1964);Physica 132A:143 (1985); I. Müller,Z. Physik 198:329 (1967); L. S. García-Colín, M. López de Haro, R. F. Rodríguez, J. Casas-Vázquez, and D. Jou,J. Stat. Phys. 37:465 (1984).
W. Israel,J. Math. Phys. 1:1163 (1963);Physica 106A:204 (1981).
J. Ehlers, inGeneral Relativity and Cosmology, Proceedings International School of Physics Enrico Fermi 1969 (Academic Press, New York, 1971).
J. M. Stewart,Non-Equilibrium Relativistic Kinetic Theory (Lecture Notes in Physics No. 10; Springer, Berlin, 1971).
S. R. de Groot, W. A. van Leeuwen, and C. G. van Weert,Relativistic Kinetic Theory (North-Holland, Amsterdam, 1980).
H. Grad,Phys. Fluids 6:147 (1963).
H. Grad,Commun. Pure Appl. Math. 2:331 (1947).
J. L. Synge,The Relativistic Gas (North-Holland, Amsterdam, 1957).
N. G. van Kampen,Physica 43:244 (1969).
F. Jüttner,Ann. Phys. (Leipzig)34:856;35:145 (1911).
C. Eckart,Phys. Rev. 58:919 (1940).
L. Landau and E. M. Lifshitz,Fluid Mechanics (Pergamon, Oxford, 1959), p. 499.
N. A. Chernikov,Phys. Lett. 5:115 (1963);Acta Phys. Polon. 27:465 (1964); B. Vignon,Ann. Inst. Henri Poincaré 10:31 (1969); C. Marle,Ann. Inst. Henri Poincaré 10:127 (1969).
H. Falk, private communication.
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van Kampen, N.G. Chapman-Enskog as an application of the method for eliminating fast variables. J Stat Phys 46, 709–727 (1987). https://doi.org/10.1007/BF01013381
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DOI: https://doi.org/10.1007/BF01013381