G (ta, q0 ; t, q) = J Dq exp(- J L(t, q, q )dt ), (2.25.10)

where

J2m(to, qo )Ato Ni=1 V2naW-1,q¿-1 )Ati and the Lagrangian, L(t,q,q), where q = àq/At, is:

2a 2 dq tJq

The path integral in Eq. 2.25.10 is consistent with the path integral directly derived from the Fokker-Planck equation (Drozdov, 1993). For higher dimensions the derivation of the path integral from stochastic differential equations is much more complicated. Interested readers may find rigorous calculations by Langouche et al. (M1982). When the functional integral jL(t, q, q)dt is minimized, it yields an Euler-Lagrange equation. Thus the Lagrangian in Eq. 2.25.12 may be used to find the most probable trajectory for particles.

2.25.4. Main results and method's of checking

Zhang (1999a,b,c) have presented a Markov stochastic process approach to the diffusion theory of CR modulation, propagation and acceleration. The CR transport equation is reformulated with the Itô stochastic differential equation. From the stochastic differential equation a Fokker-Planck equation can be derived for the probability density, which is proportional to the CR flux. The transition probability density of the Markov process is obtained as a path integral consistent with that derived in quantum mechanics. Fig. 2.25.1 shows an example of computer calculations of modulated CR spectra.

From Fig. 2.25.1 it can be seen that the three different methods - the path integral approach, stochastic process simulation and numerical method to solve the diffusion equation ('SolMod' according to Fisk, 1971) - all agree with each other. In addition to the ability to solve the CR diffusive transport equation, the two new methods provide the detailed physical processes behind their solutions (Zhang, 1999a,b).

2.26. Velocity correlation functions and CR transport (compound diffusion)

The transport of energetic charged particles in a turbulent magnetic field is often diffusive, where the time evolution of the omnidirectional particle density fo (xt, t ) is described by a diffusion equation with diffusion tensor Kj . Jokipii and Kota

(1999) consider some consequences of a particular way of looking at the diffusion. For a random, diffusive motion, the spatial diffusion tensor Kj can be related, under very broad conditions, to the velocity correlation function Kubo (1957)

in the limit that t ^ ™ . Here Ax = xj (t +1') - xj (t) is the spatial displacement of particle positions between times t and t + t'; the brackets < > denote averages over an ensemble. It was assumed that the fluctuating velocities are statistically homogeneous over the time and length scales of interest, so the velocity correlation function (vj (t(t +1) depends only on the time difference t'. For any physical random velocity (vj(0)vj (t') must go to zero at large t', and the integral in Eq. 2.26.1

approaches a constant value for t'^ ^ . The virtue of this method is that only the particle velocity needs to be considered. Forman (1977a) was the first to apply Kubo's formalism to the transport of CR. It has recently been invoked by Bieber and Matthaeus (1997), who postulated a simple exponential form for the correlation tensor (vj (0)vj (t') to infer the corresponding perpendicular diffusion coefficient k±

and effective drift, which is related to the anti-symmetric component of Kjj. Jokipii and Kota (1999) consider the special case of the perpendicular diffusion of particles tied to the turbulent magnetic field lines, which is important in understanding the transport of energetic charged particles in the Heliosphere, and for which the application of Kubo's formalism is not obvious.

According to Jokipii and Kota (1999) the most poorly-understood area of CR transport at present is the transport of particles perpendicular to the direction of the average magnetic field. This motion is owed to at least two distinct effects. Particles may scatter across field lines and the field lines may depart from the mean field owing to the random walk and mixing of field lines (Jokipii and Parker, 1969). This random walk of the field lines plays an important role in the perpendicular diffusion (see, e.g. Jokipii, 1966; Forman et al., 1974; Giacalone and Jokipii, 1999). Low rigidity particles in certain cases may be effectively tied to magnetic field lines, so it is useful to consider an idealized, but physically consistent, approximation, in which particles are assumed to be strictly tied to the field lines. The particles are assumed scatter back and forth along the field lines, in which case the particle perpendicular transports arises solely from the random walk of field lines. This can then serve as a starting point for understanding the more general problem of particle transport. This approximation has been termed compound diffusion, and has been used to discuss transport of CR in the Galaxy (e.g. Getmantsev, 1963; Lingenfelter et al., 1971; Allan, 1972).

Compound diffusion may be written as the convolution of two diffusive processes. Particles scatter back and forth and spread strictly along the field lines with a diffusion coefficient %, and the field lines, in turn, diffuse perpendicular to the mean field's in z-direction with a diffusion coefficient DL . The mean square displacement in a perpendicular direction, say x, is then proportional to the length traveled along the field line, which, from simple scaling properties, is proportional to yfKt . A quantitative calculation evaluating the convolution of the x and t motions yields

which is slower than the standard diffusion, where ^A*2^ = 2K±t, and so is fundamentally non-Markovian.

2.26.3. The Kubo formulation applied to compound diffusion

Kubo's (1957) formalism states essentially that the mean square displacement

(ax2) in a time At can be obtained from very general principles as

If At

which for At large compared with the coherence time of v* (t) yields diffusive motion with a diffusion coefficient given by

The only requirement, in addition to statistically homogeneous conditions, is that the velocity correlation function v* (o)v* (t) should vanish sufficiently fast as the time lag t increases. Under these conditions the Kubo model would always give a diffusion « At, and could not yield compound diffusion, which results in a slower transport ^ At12. There is clearly a problem with the application of Kubo's formulation to this problem. To explore this more deeply, to see where the problem lies, Jokipii and Kota (1999) have considered a simple, transparent model in which the particles propagate either forward or backward along a magnetic field line, which executes random walk about the main field direction in z. The z axis points in the direction of the mean background field; £ denotes the position, measured along the field line, and *(£) stands for the departure of the field line from the mean field. They consider particles released, in random directions, at £= 0 (x = 0) at time t = 0. The variation of the number of forward (n+) and backward

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