2ux u

If 1

c2 c2

This result coincides with that obtained for non-relativistic case (compare with Eq. 4.4.14 at Ek >> Eko = macu2 ¡2). According to Eq. 4.4.37 only 25% of the energy gain is caused by the difference in the frequencies of particle collisions with clouds; 75% of energy gain is caused by the systematical small gain energy which does not depend from 9 and which was neglected in original variant of statistical acceleration mechanism (Fermi, 1949) as well as in many subsequent papers of other authors.

Thus, the effect of a systematic small excess of energy gain over energy loss in each collision, when averaged over all possible angles between u and v, is of main importance to the statistical energy gain by a particle - not only in the low-energy range (see Section 4.4.1) but also at relativistic energies.

Now let the particle energy change in time be found:

c2ll 3

where the frequency of particle collisions with clouds v(p< = w(p<!A(p< was determined by Eq. 4.4.33 and 4.4.35. Here w(p< is the velocity of particle relative to the cloud, and A(p< is the particle transport path. In Eq. 4.4.38 we show separately two parts of the energy gain (the same 25% and 75%) caused correspondingly by differences in frequencies of collisions and by small but systematic energy gain. Integrating over Eq. 4.4.38 with the initial condition E = Ei at t = 0 , we obtain for relativistic particles (v = c):

'Ei exp f8u2tI

where the last expression is valid if the parameter of acceleration a = 8u2¡3cA does not change with time during the particle's acceleration. Using the particle distribution determined by Eq. 1.1.6 over the age t from the acceleration onset and including the relations t _ ^ln(E/E,),

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