As we have seen, the asymptotic reduction of the original CR propagation problem, given by Equation (9), to its isotropic part cannot proceed to higher orders of approximation using a simple asymptotic series in Equation (10) and requires a multi-time asymptotic expansion. In the Chapman–Enskog method, the operator 





 is expanded instead. Its purpose is to avoid unwanted higher time derivatives to appear in higher orders of approximation. This is very similar to, e.g., a secular growth in perturbed oscillations of dynamical systems. To eliminate the secular terms, one seeks to alter (also expand in a small parameter) the frequency of the zero-order motion, which is similar to the 





 expansion. One example of such an approach may be found in a derivation of hydrodynamic equations for strongly collisional, but magnetized plasmas, starting from the Boltzmann equation (Mikhailovsky 1967). The classical monograph by Chapman & Cowling (1991; Ch.VIII) gives another example of a subdivision of the 





 operator for solving the transport problem in a non-uniform gas-mixture. Expanding 





 operators eliminates secular terms, such as the telegraph term. Perhaps more customary today, and equivalently, is to introduce a hierarchy of formally independent time variables (e.g., Nayfeh 1981) 





, so that
14





Instead of Equation (13), from Equation (9), we have
15





where the conditions 





 are implied. The solution of this equation should be sought in the following form
16





where 





 and 





 are chosen to satisfy, respectively, the following two equations.
17





and
18





The solution for 





 is as follows
19





and it can be evaluated for arbitrary n by expanding both sides of Equation (17) in a series of eigenfunctions of the diffusion operator on its lhs:





For D = 1, for example, 





 are the Legendre polynomials with 





, 





. The time dependent coefficients 





 are determined by the initial values of 





 (the anisotropic part of the initial CR distribution) and the rhs of Equation (17), that depends on 





, obtained at the preceding step. It is seen, however, that 





 exponentially decay in time for 





 and we may ignore them3

3
In fact, we must do so because our asymptotic method has a power accuracy in 





, but not the exponential accuracy.
 because we are primarily interested in evolving the system over time 





 and even longer. Starting from n = 0 and using Equation (15), for the slowly varying part of f, we have
20





The solubility condition for 





 (obtained by integrating both sides of Equation (15) in μ) also gives a trivial result
21





so the last two conditions are consistent with the suggested decomposition in Equation (16), since from Equation (18) with n = 1, we have
22





and, thus both 





 and 





 are, indeed, independent of 





 and 





. We have introduced the function 





 here by the following two relations
23





The solubility condition for 





 yields the nontrivial and well-known (e.g., Jokipii 1966) result, which is actually the leading term of the 





 expansion in 







24





where





The solubility conditions for 





, 





 will generate the higher order terms of our expansion, which, after some algebra, can be manipulated into the following expressions for the third and fourth orders of approximation
25







26





We have denoted





and 





. The pitch-angle diffusion coefficient 





 and magnetic focusing σ are considered z-independent for simplicity, a limitation that can be easily relaxed by rearranging the operators containing 





 in Equation (26). We can proceed to higher orders of approximation ad infinitum since terms containing 





 can be expressed through 





. According to Equations (20)–(21), of interest is the evolution of 





 on the timescales 





 or 





; thus, as we already mentioned, the contributions of 





 to all of the solubility conditions, similar to those given by Equations (24)–(26), have to be dropped (because they become exponentially small) and only 





-contributions should be retained. Using Equations (20)–(21) and (24)–(26) to form the combinations 





 and summing up both sides, on the lhs of the resulting equation, we simply obtain 





 (see Equation (14)). Therefore, the evolution of 





 up to the fourth order in ϵ takes the following form.
27





where 





, 





, and
28
Citation: Nayfeh 1981
Citation function: Uses; Uses