1. SUBDIFFUSION OF BEAMS THROUGH INTERPLANETARY AND INTERSTELLAR MEDIA Aleksander Stanislavsky Institute of Radio Astronomy, 4 Chervonopraporna St., Kharkov 61002, Ukraine alexstan@ira.kharkov.ua

2. Introduction Fluctuations of the beam propagation direction in a randomly inhomogeneous medium are frequently observed in nature. Examples include random refraction of radio waves in the ionosphere and solar corona, stellar scintillation due to atmospheric inhomogeneities, and other phenomena. The propagation of a beam (of light, radio waves, or sound) in such media can be described as a normal diffusion process. One extraordinary property predicted for random media – and later revealed – is the Anderson localization. It brings normal diffusion to a complete halt. Fluctuations of the beam propagation direction in a randomly inhomogeneous medium are frequently observed in nature. Examples include random refraction of radio waves in the ionosphere and solar corona, stellar scintillation due to atmospheric inhomogeneities, and other phenomena. The propagation of a beam (of light, radio waves, or sound) in such media can be described as a normal diffusion process. One extraordinary property predicted for random media – and later revealed – is the Anderson localization. It brings normal diffusion to a complete halt.

3. Localization of waves in a disordered media Anderson localization of waves in disordered systems originates from interferences in multiple elastic scattering. The light source is denoted by a star symbol and the spheres denote of scattering elements. When two waves propagating in opposite directions along a closed path are in phase, the resultant wave is more likely to return to the starting point (to A) than propagate in other directions.

4. At strong enough scattering the system makes a phase transition in a localized regime (D. Wiersma et al,1997, 2000). This transition can best be observed in the transmission properties of the system. In the localized state the transmission coefficient decreases exponentially instead of linearly with the thickness of a simple. This feature makes waves in strongly disordered media a very interesting system. The properties of a randomly inhomogeneous medium vary not only from point to point, but also with time. Consequently, localization may take place both at random locations and at random times. Random localization affects diffusive light propagation in a random medium.

5. Problem Formulation Suppose that the medium is statistically homogeneous and isotropic. Then, a beam propagating through the medium is deflected at random. Localization implies that the beam is trapped in some region. Since the trapped beam returns to the point where it was trapped, its propagation is "frozen" for some time. After that, a randomly deflected beam leaves the legion and propagates further until it is trapped in another region (or at a point) and the localization cycle repeats. The randomly winding beam path due to inhomogeneities is responsible for the random refraction analyzed in this study.

6. The angle of deviation of a beam from its initial direction is characterized by a probability density, where is the path travelled by the beam. Let us derive an integro-differential equation for the probability density. The angle of deviation of a beam from its initial direction is characterized by a probability density, where is the path travelled by the beam. Let us derive an integro-differential equation for the probability density.

7. Continuous time random walks The random walks analysed here consist of random angle jumps The random walks analysed here consist of random angle jumps at points separated by segments of random length.Let be independent identically distributed variables governed by a  -stable distribution. Assume that the angle jumps are independent random variables belonging to the Gaussian probability distribution. at points separated by segments of random length.Let be independent identically distributed variables governed by a  -stable distribution. Assume that the angle jumps are independent random variables belonging to the Gaussian probability distribution.

8. The  -stable distributions possess the following similarity property:, where means that random values have similar distributions. Such a random variable is characterized by the probability distribution with the Laplace transform: where,,. The total path length is the sum of all. If is the number jumps, then the beam position is. On account of convergence of distributions we can definitely pass from the discrete model to a continuous limit (Meerschaert et al, 2004). where,,. The total path length is the sum of all. If is the number jumps, then the beam position is. On account of convergence of distributions we can definitely pass from the discrete model to a continuous limit (Meerschaert et al, 2004).

9. Subdiffusion of beams Both and are Markov processes. However, since the latter is the directed process to the former, the resultant process may not preserve the Markov property (Feller, 1964). This leads to the subdiffusion equation where is a generalized diffusion coefficient, and is the gamma function.

10. Its solution is where is the solution of rotational Brownian motion, and the function takes the form The subdiffusion equation yields the mean where is the Mittag-Leffler function. function. At large, all beam directions are equiprobable. At large, all beam directions are equiprobable.

11. Applications Following the method developed by Chernov (1953), Following the method developed by Chernov (1953), one can find the mean square of the distance from the starting point to the observation point reached by the beam that has traveled an intricate path of length one can find the mean square of the distance from the starting point to the observation point reached by the beam that has traveled an intricate path of length through the medium: through the medium: If, then If, then

12. If the axis of a polar coordinate system is aligned with the initial beam direction, then the mean square of the distance passed by the beam along this axis is given by the formula If, then

13. Now, the mean square deviation of the beam from its initial direction can be calculated by combining the above-obtained expressions: If,then a generalized ``3/2 law'' is obtained The case of corresponds to normal diffusion without wave localization.

14. Brief concluding remarks The experimental deviations from the ``3/2 law'' (more precisely, from an exponent of 3/2 in the classical power law) were mentioned in the paper of Kolchinskii (1952). However, the author preferred to connect their presence with systematic measurement errors, probably because of the lack of plausible interpretation. This problem can be revisited in view of the results obtained in this study. The application of this method to the diagnostics of interplanetary and interstellar turbulent media may be useful for understanding some astrophysical processes in disordered media.