( n- ) moving particles as a function of time t, and position along the field line £ is governed by the pair of equations:

where t represents the average time of scattering, q+ and q- are the number of particles released in positive and negative directions, respectively.

Obviously, the velocity in the x direction is ± v(dx/d£) depending on whether the particle happens to move forward or backward along the field line. Thus to obtain the velocity correlation (vx (o)vx (t)), the mean velocity (vn+ - vn-) along the field is to be averaged over position, with the inclusion of the actual orientation of the field line

where subscripts imply the position in Since the source q+ accounts for positive initial speed, whilst q- corresponds to negative intial speed, n+ and n- can be taken as the solutions for sources q+ = 1/2 and q- = -1/2. This ensures that the initial speed vx (o) is properly taken into account. Instead of considering the directly the velocity correlation (vx (o)vx (t)) Jokipii and Kota (1999) consider its Laplace transform

They note that Lxx(s = 0) yields exactly the corresponding perpendicular diffusion coefficient Kxx, whilst the behavior of Lxx at small s values brings information on the behavior of (vx (o)vx (t)) for large times (t >> t).

Jokipii and Kota (1999) adopt the technique of Fourier and Laplace transforms (Fedorov and Shakhov, 1993; Kota, 1994). First, taking the Fourier transform of Eq. 2.26.5 and Eq. 2.26.6, and Laplace transforming the resulting pair of equations yields a solution for Lxx (s ) :

Inspection of Eq. 2.26.9 shows that Lxx(o) = 0, unless the integral over which is related to the random walk of field lines, is infinite (this would be the case only if the field had a nonzero regular component in the x direction). Since the Laplace transform at s = 0 is identical to Kxx, the derivation above demonstrates that Kubo's theorem yields precisely zero perpendicular diffusion coefficient Kxx for compound diffusion. By the opinion of Jokipii and Kota (1999) this result could intuitively be anticipated, since compound diffusion produces slower than x-t diffusion.

A further study of Eq. 2.26.9 reveals the character of (vx (o)vx (t)) in more detail. First Jokipii and Kota (1999) notice that, for small values of s, ko ~ 0, and the integral over £ in Eq. 2.26.9 gives the power of field fluctuations at zero wavenumber, which is equivalent to 2 DL , where DL is the diffusion coefficient of field line random walk (Jokipii, 1966). For small values of s, Lxx ~ vDL(st)12 , and Eq. 2.26.9 implies that for large values of t

The velocity correlation function has a long negative tail to balance the positive values at smaller t, and to give an exactly vanishing integral in Eq. 2.26.4. This long term behavior could be obtained directly from considering the solutions for n+ and n_ of Fisk and Axford (1969) in the t >>t limit, when the exact solutions can be approximated by diffusive time profiles.

Thus Jokipii and Kota (1999) find that, in a broader sense, compound diffusion fits into Kubo's theory. The velocity correlation function (vx(0)vx(t) exhibits a long-term anticorrelation, causing the diffusion coefficient Kxx (i.e. the integral in Eq. 2.26.4) to vanish. This is connected with the non-Markov nature of the compound diffusion. At this point it is of interest to establish the connection between the present discussion and some current ideas in time-series analysis. The fact that the mean square displacement ^Ax2^ increases as At5 means that the motion is non-Markov. According to opinion of Jokipii and Kota (1999), the case in which ^Ax2^ x At2H (0 < H < 1) has been given the name fractional Brownian motion, where H is the Hurst exponent (Mandelbrot and Van Ness, 1968). The case of compound diffusion corresponds to a Hurst exponent H = 0.25. It may be shown (see Section 9.4 in Feder, M1988), that if H > 0.5 the process exhibits long-term positive correlation and conversely, if H < 0.5 corresponds to long-term anti-correlation of the process. Clearly, then, the case of no correlation requires H = 0.5. This agrees with the determination using Laplace transforms, described previously. Now, physically, it is expected that particles in a turbulent magnetic field will loose correlation and it will be retrieve the standard form (a*2) x t. But this cannot occur if particles are strictly tied to field lines.

Jokipii and Kota (1999) considered compound diffusion, which is a non-Marko diffusion leading to Ax2 ^ At12. This idealized but valid motion is seemingly in contradiction with Kubo's theory, which yields Ax2 « At. They have shown that compound diffusion fits into the general theory in a broader sense and determined the Laplace transform of the velocity correlation, and showed that the diffusion coefficient, as defined by Eq. 2.26.1, turns out to vanish. A study of the Laplace transform revealed, furthermore, that the velocity correlation has a negative non-exponential tail, (v* (0)v* (t))x-tindicating a long-term anticorrelation. The

A*2 «: At12 behavior of the compound diffusion could also be recovered from Kubo's formalism. The compound diffusion may serve as a starting point for understanding the perpendicular transport of low rigidity particles. The question is how to proceed from this picture to a model including some scattering across field lines. A small amount of cross-field scattering can be amplified by the subsequent mixing of field lines; originally nearby field lines may separate to great distances. The time scales of these processes may be large for low rigidity particles. In this case the long non-exponential tail of the velocity correlation, which is a result of the long-term anticorrelation, may be of importance; the velocity correlation function may considerably differ from the simple exponential decay postulated by Bieber and Matthaeus, (1997). These questions, in the opinion of Jokipii and Kota (1999), need further exploration. They also point out that consideration of temporally varying magnetic fields suggests that the conclusions derived here apply also to this situation.

2.27. The BGK Boltzmann equation and anisotropic diffusion

Early work by Parker (1965) and Axford (1965) derived the form of the diffusion tensor for CR in a random magnetic field for the case of isotropic scattering. Forman et al. (1974) used quasi-linear theory in slab turbulence to determine the diffusion coefficients parallel Kjj and perpendicular kl to the mean magnetic field Bc, as well as the anti-symmetric component of the diffusion tensor ka , associated with particle drifts, for the case where the distribution function could be expanded in spherical harmonics. Jokipii (1971) and Hasselmann and Wibberenz (1970), pointed out that the detailed dependence of the pitch-angle diffusion coefficient Dmon j is important in determining k//. Webb et al. (2001)

study a model of CR diffusion based on a gyro-phase, and pitch-angle dependent BGK Boltzmann model, involving two collision time scales t__ and t// associated with scattering perpendicular and parallel to the background magnetic field Bo . The time scale t// describes the ironing out of gyro-phase anisotropies, and the relaxation of the full gyro-phase distribution f to the gyroaveraged distribution fo . The time scale t± determines the diffusion coefficient k± , perpendicular to the mean magnetic field, and the corresponding anti-symmetric diffusion coefficient ka associated with particle drifts. The time scale t// describes the relaxation of the pitch-angle distribution fo to the isotropic distribution Fo, and determines the parallel diffusion coefficient k// . The Green's function solution of the model equation is obtained, for the case of delta function initial data in position, pitch-angle, and gyro-phase, in terms of Fourier-Laplace transforms. The solutions are used to discuss non-diffusive and diffusive particle transport. The gyro-phase dependent solutions exhibit cyclotron resonant behavior, modified by resonance broadening due to t± . Below, in Sections 2.27.2_2.27.7, the model of Webb et al. (2001) will be considered in details.

According to Webb et al. (2001) the BGK Boltzmann equation for the momentum-space distribution function f (r, p, t), for particles with momentum p, (or velocity v), at position r at time t, in a uniform background magnetic field Bo = {0,0,Bo} along the z-axis, may be written in the form:

f + v-Vf = _l_f°_ fo _Fo , (2.27.1) dt dp t_|_ T//

where

denote the gyro-phase averaged distribution function ( fo ), and the isotropic component of the distribution function (Fo) in momentum space, and n = cosd is the pitch-angle cosine. The gyro-phase derivative term -Q(//30), on the left hand side of Eq. 2.27.1, is the Lorentz force term, where Q = qBolmc is the particle gyro-frequency, and m is the relativistic particle mass. Note that (v, 9,0) are spherical polar coordinates for the velocity, where the polar axis is along Bo . Kota (1993) used a model similar to Eq. 2.27.1, except that he used a pitch-angle and gyro-phase diffusion term for the collision term.

2.27.3. The diffusion approximation

Following the approach of Kota (1993), Webb et al. (2001) expand the distribution function in the series:

where /-n = /* . Multiplying the Boltzmann Eq. 2.27.1 by exp((m0), and integrating over the gyro-phase 0 from 0 = 0 to 0 = 2n, yields the moment equations:

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