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SUBDIFFUSION OF BEAMS
THROUGH INTERPLANETARY
AND INTERSTELLAR MEDIA
Aleksander Stanislavsky
Institute of Radio Astronomy,
4 Chervonopraporna St.,
Kharkov 61002, Ukraine
[email protected]
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.
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.
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.
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.
The angle  of deviation of a beam from its
initial direction is characterized by a probability
density W ( , ) , where  is the path travelled
by the beam. Let us derive an integro-differential
equation for the probability density.
Continuous time random walks
The random walks analysed here
consist of random angle jumps  i
at points separated by segments of
random length  i .Let  i 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.
The -stable distributions possess the following
similarity property: t  X + s  X  (t + s)  X , where
means d
that random values have similar
distributions. Such a random variable is
characterized by the probability distribution with
the Laplace transform:
1/
d
1/
1
1/
2
3

 ( s)   exp(  sx )df ( x )  exp( ( Ax) ),
0
where x  0 , A  0 , 0    1 . The total path length
is the sum of all  i . If N is the number jumps,
N
 N  i 1 i
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).


Subdiffusion of beams
 i
Both  i 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
W ( ,  )  W ( ,0) 

D
 
W ( , ' ) 
 1

 sin 
   '  d '
( ) sin   


0
where D is a generalized diffusion coefficient,
and ( ) is the gamma function.
Its solution is
W ( ,  ) 


F
(
z
)
W
(

,

z )dz
1
 
0
where W1 is the solution of rotational Brownian motion,
and the function F (z ) takes the form
( z)
F ( z )  
.
k 0 k! (1    k )

k
The subdiffusion equation yields the mean
cos  E ( 2 D  ),
where

E ( x )   x n / (1  n )
is the Mittag-Leffler
function.
n 0
At large  , all beam directions are equiprobable.
Applications
Following the method developed by Chernov (1953),
one can find the mean square of the distance r from the
starting point to the observation point reached by the
beam that has traveled an intricate path of length 
through the medium:


1
2


r 

1

E
(

2
D

) .

2
D(  1) 2 D
If D  1 , then



1
2
D

2
2
r  2 
 (2  1)  (3  1) 
.


v
z axis of a polar coordinate
If the
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



1

1
2


z 

1  E ( 6 D ) .

3D  (  1) 6 D

If D  1 , then



1
6
D

2
2
z  2 
 ( 2  1)  (3  1) 
.


Now, the mean square deviation of the beam from
its initial direction can be calculated by
combining the above-obtained expressions:

2

 2  r2  z2 

3D(  1)
1
1






1

E
(

2
D

)

1

E
(

6
D

) .


2
2
2D
18D
If D  1 ,then a generalized ``3/2 law'' is obtained
2 
2 2
D1 / 2 3 / 2 .
(3  1)
The case of   1 corresponds to normal diffusion
without wave localization.
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.