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Astron. Astrophys. 327, 432-440 (1997)

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3. Continuous time random walks

We do not aim to describe the microscopic dynamics of particles moving in a turbulent magnetic field in a realistic way, but rather look for a model which is able to reproduce their transport characteristics and gives the density profile. In terms of the propagator [FORMULA], the density is given by Eq. (2), once the rate [FORMULA] at which particles are injected is specified. This can then be used to compute, for example, the synchrotron radiation from electrons injected at a plane surface (such as a spiral galaxy) and subsequently transported outwards.

The method we adopt to model P is that of continuous time random walks (CTRW). In this, the transport of a particle is modelled as a succession of instantaneous jumps of arbitrary length, separated by pauses of arbitrary duration. Each of these steps is independent of the previous steps. We denote by [FORMULA] the probability distribution function (PDF) of a jump described by [FORMULA], and by [FORMULA] the PDF of a pause of duration t, and use their Fourier and Laplace transforms:

[EQUATION]

The type of anomalous transport characterised by [FORMULA] and discussed above arises from the asymptotic behaviour of [FORMULA] as [FORMULA] and [FORMULA]. This corresponds to the [FORMULA], [FORMULA] asymptotic behaviour of [FORMULA], for which the expansions of [FORMULA] and [FORMULA] around 0 are required. To keep the treatment general, we consider first the motion of a particle in a space of dimension d. The functions [FORMULA] and [FORMULA] are then prescribed as

[EQUATION]

with [FORMULA], and [FORMULA], [FORMULA] positive constants. In Eq. (9), we have assumed that [FORMULA] depends only on [FORMULA], i.e., that there is no preferred direction for a particle jump. The particular values [FORMULA], [FORMULA] yield a diffusive process with a Gaussian propagator, whereas the case [FORMULA], leads to a random walk with an infinite second moment [FORMULA], corresponding to Lévy flights.

In a classical paper, Montroll and Weiss (1965 ) have shown that the Fourier-Laplace transform of [FORMULA] has the following form

[EQUATION]

Inserting the expansions Eq. (9) we find that for large [FORMULA] and t

[EQUATION]

and the transport coefficient is [FORMULA]. In the following, we find it convenient for the inversion of the Fourier transforms to choose [FORMULA], and describe all transport regimes using the parameter range [FORMULA]:

[EQUATION]

for small s and [FORMULA]. The choice [FORMULA] has the same asymptotic behaviour as the function [FORMULA], i.e., [FORMULA], which corresponds to scattering after a fixed time interval, yielding diffusion. However, for [FORMULA],2, we have

[EQUATION]

for [FORMULA]. This is a long-tailed distribution function. Its slow decrease at large t leads to the same kind of memory effect built in to the non-Markovian transport equation (6) constructed by Chuvilgin & Ptuskin (1993 ) and Balescu (1995 ) for the case [FORMULA]. We do not consider [FORMULA], since in this case the quantity [FORMULA] increases without limit for large t, and would at some stage exceed the actual physical particle speed.

From the expansions (13) the asymptotic behaviour of the propagator at large [FORMULA] and t can be found by an inverse Fourier transformation followed by an inverse Laplace transformation, which is performed approximately using the method of steepest descents. The resulting expression can be used for all values of [FORMULA] and t once the normalisation has been corrected. It represents a convenient one-parameter model of particle transport which exhibits the anomalous behaviour in which we are interested. The details of the derivation are given in Appendix A. Here we simply present the results, expressed in terms of a dimensionless similarity variable defined according to

[EQUATION]

Using this variable, the propagator separates:

[EQUATION]

and we find

[EQUATION]

where the (positive) constants [FORMULA] and A are given in the appendix (Eqs. A3 and A4). These propagators are normalised such that particle number is conserved: [FORMULA]. They display at all t the behaviour required of the asymptotic dependence of the variance in anomalous transport. In particular, they possess for all t the property

[EQUATION]

For one-dimensional propagation ([FORMULA]), which describes, for example, the transport of particles in x away from a uniform source in the y -z plane, we have for the standard diffusive case ([FORMULA]) the well-known form

[EQUATION]

with [FORMULA] and [FORMULA]. In the [FORMULA] sub-diffusive case, which can be considered as a double diffusion, we find:

[EQUATION]

which agrees with the propagator for [FORMULA] given in Eq. (5) and derived by Duffy et al. (1995 ), with [FORMULA] and [FORMULA]. (Similarly, the case [FORMULA] reproduces the propagator (3) of Rax & White 1992 ). In analogy with the formulation of Eq. (4), it is possible to consider other values of [FORMULA] as arising from a convolution of a propagator describing motion along the field line together with one describing the wandering of the field. Using the method of steepest descents, it is straightforward to show that [FORMULA] has the following convolution property:

[EQUATION]

(note that the normalisation is preserved) so that the combined transport process can always be described using a value of [FORMULA] between 0 and 2.

To illustrate the properties of these propagators, they are plotted for three different values of Fig.1.

[FIGURE] Fig. 1. The dimensionless propagator (Eq. 16) for one-dimensional transport in three different regimes: standard diffusion ([FORMULA], solid line), sub-diffusion ([FORMULA] dotted line) and supra-diffusion ([FORMULA] dashed line), as a function of the dimensionless similarity variable [FORMULA], which, for fixed time, is proportional to distance (see the definition in Eq. 15).

The diffusive propagator is simply a gaussian curve given by Eq. (19) according to which the probability of finding a particle at a particular distance drops off smoothly with increasing distance from the point (plane) of injection. The sub-diffusive propagator, on the other hand, falls off much more steeply initially (it has an integrable singularity at the point of injection), but at larger distances decreases more slowly than diffusion. Acting on an ensemble of particles, sub-diffusive transport tends to confine some of them close to the injection point, but propagates others very rapidly to large distance. This is a result of the larger spread in position of sub-diffusing particles of a given age, compared with diffusing ones. Thus, despite the fact that sub-diffusion produces, on average, slower transport, a minority of particles experiences comparatively rapid transport. Supra-diffusion displays the opposite behaviour and is similar to ballistic transport, or propagation at constant speed without scattering. The propagator is strongly peaked around the value [FORMULA]. Very few particles are to be found lingering close to the point of injection, and very few escape to large distance.

Eq. (17) is the basic result of this section. We now turn to the question of the observable consequences of these propagators.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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