x {(r,t;ri,ti ))±rM - (V±rMBfr (r,t;ii,ti^p|)}(ii,pi,ti), (2.8.17)

Averaging of the third term of the right hand side of Eq. 2.8.12 to within the accuracy of second-order terms in the random field leads to the relationship

(#1g(r, t )Hir(r, t )f [Hi] = Bfr(r, t;r, t )F (r, p, t). (2.8.18)

Taking Eq. 2.8.17 and Eq. 2.8.14 into account we obtain:

\dt~f [Hiy = V/'VaF (r, P, t)-V//Aa?(n)^ dt1 ' dridpiGpp1 (t;rl, ti )vL//

x {{ (r, t;ri, ti )V±ri^-(v±ri^B^w(r, t; rb ti ^pil} (ri, pi, ti), (2.8.19)


Averaging of the third and fourth terms on the left side of Eq. 2.8.11 is performed similarly:

t f [H1] = -V// pij dt1 J dr1<p1Gpp1 (r, t;rb t1))//

x \VLrßVLr^Bßfi (r't; r1' t1 fäPu - (rß%(r,t;r1,t1 )V±r^)(rbpb^

dt 2 dt

We shall make use of the simplest model of scattering of particles by statistical inhomogeneities of the magnetic field (Dolginov and Toptygin, 1966a), in which one can neglect the influence of the large scale field on scattering of particles at scales of the order of the correlation radius Lc of the very small scale field, to average the collision integral St/. In this case

where A(p) is the transport mean free path of a particle with respect to scattering by very small scale inhomogeneities of the magnetic field. Taking Eq. 2.8.19-2.8.21 into account, we write the kinetic Eq. 2.8.11 averaged over the large-scale random field:

Jt- + VrnV// - St jF(r, p,t) = Vir«*WVirlF(r - P, P1,t - t)

where the operators Kn and D are defined by the relationships

If the random field described by the tensor Bai(p,r) is delta-correlated in time, Eq. 2.8.22 takes the form corresponding to the Fokker-Planck kinetic equation:

+ Vrn V// - St^F(r,p, t) = ViraKioiVirAF(r,p, t) + V//Op2±DOF(r,p, t), (2.8.25)


In many cases the average distribution function and its derivatives with respect to the momentum vary slowly at the characteristic spatial and temporal scales of variation of the random field, one consequently can remove the distribution function from under the integral sign on the right hand side of Eq. 2.8.22 and switch to the Fokker-Planck Eq. 2.8.25. In this case

K±aA = V// Jdz\dpdP1Gpp1 (p,t)v1//5oa(P,t), (2.8.27)

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