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Different CR acceleration mechanisms in space plasmas were partly reviewed in the books Alfven (M1950), Dorman (M1957, M1963a,b, M1972b, M1975a, Ml 978), Ginzburg and Syrovatsky (M1963), Parker (M1963), Pikelner (M1966), Rossi (M1966), Dorman and Miroshnichenko (M1968), Hayakawa (M1969), Tsytovich (M1971), Khristiansen (M1974), Arons et al., eds. (M1979), Melrose (M1980a,b), Priest (M1982), Toptygin (M1983), Berezhko et al. (M1988), Berezinsky et al. (M1990), Zank and Gaisser, eds. (M1992), Benz (M1993), Sturrock (M1994), Ramaty et al., eds. (M1996), Priest and Forbes (m2000), Miroshnichenko (M2001), Schlickeiser (M2001), and in review papers Dorman and Katz (1977), Syrovatsky (1981), Axford (1987), Debrunner (1987), De Jager (1987), Galeev et al. (1987), Ginzburg (1987), Ramaty (1987), Völk (1987), Dorman and Venkatesan (1993), Biermann (1993), Mandzhavidze (1993), Berezhko (1997, 2001), Cane (1997), Baring (1999), Cliver (1999), Kirk and Duffy (1999), Akasofu (2001), Malkov and Drury (2001), Mazur (2001), Ostrowski (2001), Aschwanden (2002), Cohen (2003), Lin (2003), Moskalenko (2003), Ryan (2005), Kahler et al. (2005), Ptuskin (2005).

4.2. The Fermi mechanism of statistical acceleration

According to Fermi (1949) at each collision of a charged particle moving with velocity v, with magnetic cloud moves with velocity u, changes its energy according to the relation where the upper sign is for head-on collisions and bottom sign for overtaking collisions (see Fig. 4.2.1).

Fig. 4.2.1. Charged particle interaction with a moving magnetic cloud: a - the case in which the cloud moves against the particle (head-on collision), b - the case in which the cloud moves in the same direction as the particle (overtaking collision). According to Fermi (1949).

Fig. 4.2.1. Charged particle interaction with a moving magnetic cloud: a - the case in which the cloud moves against the particle (head-on collision), b - the case in which the cloud moves in the same direction as the particle (overtaking collision). According to Fermi (1949).

Therefore, according to Fermi (1949), in a head-on collision we shall have a relative gain energy of 2uvjc2, and in an overtaking collision the same relative loss energy of 2uv/c2 . If X is the mean free path for particle collisions with magnetic clouds, the corresponding frequencies v± for collisions will be v±=(v ± u)2A; v++v-= v/A . (4.2.2)

The average change energy per unit of time will be (including Eq. 4.2.1 and Eq. 4.2.2)

is the so called parameter of acceleration. From Eq. 4.2.3 it follows that if particle start to accelerate at t = 0 from the initial energy , their energy at moment t will be (if we neglect by energy losses on ionization and other processes):

and in case a = const it will be

Let us suppose (following to Fermi, 1949), that the process of particle acceleration in some volume is stationary, at least for a time much larger than the average time of particles time life t in the acceleration volume and that particles start to accelerate with the same probability in any time between 0 and t. In this case the particle distribution over the total time t of acceleration will be

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