The Eq. 4.11.41 with the diffusion coefficient determined by Eq. 4.11.42 and the Eq. 4.11.43 constitute a self-consistent set for determining the distribution function of accelerated particles f (p) and the wave vector B(k ). Solution of this set by the method of successive approximation gives a spectral function B(k ) of the form
B1H;2m 642vnk 2
where B1 is a constant. In accordance with Eq. 4.11.44, the Eq. 4.11.41 for the power exponent of magnetic inhomogeneity spectrum v = 2 gives an exponential spectrum of accelerated particles in the non-relativistic case. In fact, as was shown in (Dorman and Katz, 1977), if the initial function fo (p )-f (p, t = 0)
differs from zero in some region p < p0 << 1 (in units macc), the asymptotic f (p, t) at great t and p >> p0 is of the following form:
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