Mathematical Methods in Kinetic Theory by C. Cercignani

By C. Cercignani

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Grad in 1949, that, in the limit when N goes to infinity and u goes to 0 in such a way that Nu 2 tends to a finite number (the so-called Boltzmann-Grad limit), a one-particle distribution obtained from the Liouville equation should tend, in a suitable sense, to a solution of the Boltzmann equation. We remark that Nu 2 (multiplied by a typical molecular velocity and divided by the volume of the region occupied by the gas) is the order of magnitude of the right-hand side of Eq. 11). In order to give an idea of the orders of magnitude involved in a typical case, we note that Nu 2 = 104 cm 2 for N = 1020 and u = 10-8 em, while Nu 3 (a parameter measuring the importance of the terms that we have neglected) equals 10- 10 m 3 • In 1972 a "proof" of the conjecture was given (ref.

1. 7). However, we can also change the names of the integration variables and call ~ and ~ 1 what we called f,' and f,'1 before, as a consequence, and because of the involutory character of the transformation (1. 7), we can consistently call f,' and ~'1 what we called ; and ~ 1 before, and write Eq. 12) where B(O, V) is not affected by the change, since Eq. 14) of Chapter I implies V' = V. We can rewrite Eq. 13) This equation is identical to Eq. 4) except for a minus sign and for having q>(f,') in place of q>(f,).

12) of Chapter I as a linear transformation (with coefficients depending on a) of six variables into another six variables, comparison of Eqs. 12) of Chapter I and (1. 8) where I is the identity matrix and A · A denotes the usual rows-by-columns product. In accord with Binet's theorem on the determinant of a product, Eq. 9) 42 Basic Properties [Ch. 10) 1 the vertical bars denoting absolute value. But det A is precisely the Jacobian of both the direct and inverse transformation; then Eq. 10) implies that d~ d~ 1 = d~' d~'1 and Eq.

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