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H a w k e r , a n d W . C . S c h i e v e , " I n f l u e n c e of I n i t i a l C o n d i t i o n s on N o n - M a r k o f f i a n Effects in Kinetic T h e o r y " , Fourth C a n a d i a n Symposium on T h e o r e t i c a l C h e m i s t r y , V a n c o u v e r , B . C . , 1971. 34 SUBDYNAMICS IN KINETIC THEORY 16. R. 29 17. R. 18. P. 19. R. 20. R. Balescu, L. Brenig and J. Wallenborn, Physica 56 # (1971). Balescu, Physica 27, 693 (196« . A. M. Dirac, Rev. Mod. Phys. 21 (1949) . Balescu, R. Kotera, Physica 33, 558 (1967).
J. P. Guiraud, J. de Mecanique, 9^, 443 (1970). 15. T. Carleman, Problemes mathematiques dans la theorie cinltique des gaz, Almqvist and Wiksells, Uppsala,1957, 16. H. , Vol. 17, Amer. Math. , 1965. 52 SOLUTIONS OF BOLTZMANN'S EQUATIONS 17. S. Tsuge, Phys. Letters 33A, 145 (1970). 18. H. Grad, "Levels of Description in Statistical Mechanics and Thermodynamics," in Delaware Seminar on the Foundations of Physics, Springer Verlag, 1967. 19. H. Grad, J. Chem. , 33^, 1342 (1960). 20. E. Hopf, J. Rational Mech.
For example, a common method of proving that a series is asymptotic to a solution of a differential equation is to construct (by iteration) a convergent sequence which at the n'th iteration agrees (if expanded) with n terms of the expansion; both the series and the iterative sequence are equally asymptotic. This is the key to the distinction between the Hilbert and Chapman-Enskog expansions, each of which is asymptotic to a true solution of the Boltzmann equation. Hilbert expands the solution as a power series in a parameter, ε.