## 3. Continuous time random walksWe 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 , the density is given by Eq. (2), once the rate 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 The type of anomalous transport characterised by
and discussed above arises from the asymptotic
behaviour of as and
. This corresponds to the
, asymptotic behaviour of
, for which the expansions of
and around 0 are
required. To keep the treatment general, we consider first the motion
of a particle in a space of dimension with , and , positive constants. In Eq. (9), we have assumed that depends only on , i.e., that there is no preferred direction for a particle jump. The particular values , yield a diffusive process with a Gaussian propagator, whereas the case , leads to a random walk with an infinite second moment , corresponding to Lévy flights. In a classical paper, Montroll and Weiss (1965 ) have shown that the Fourier-Laplace transform of has the following form Inserting the expansions Eq. (9) we find that for large
and and the transport coefficient is . In the following, we find it convenient for the inversion of the Fourier transforms to choose , and describe all transport regimes using the parameter range : for small for . This is a long-tailed distribution
function. Its slow decrease at large From the expansions (13) the asymptotic behaviour of the propagator
at large and Using this variable, the propagator separates: where the (positive) constants and For one-dimensional propagation (), which
describes, for example, the transport of particles in with and . In the sub-diffusive case, which can be considered as a double diffusion, we find: which agrees with the propagator for given in Eq. (5) and derived by Duffy et al. (1995 ), with and . (Similarly, the case reproduces the propagator (3) of Rax & White 1992 ). In analogy with the formulation of Eq. (4), it is possible to consider other values of 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 has the following convolution property: (note that the normalisation is preserved) so that the combined transport process can always be described using a value of between 0 and 2. To illustrate the properties of these propagators, they are plotted for three different values of Fig.1.
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 . 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. © European Southern Observatory (ESO) 1997 Online publication: April 8, 1998 |