*Astron. Astrophys. 331, 1103-1107 (1998)*
## 3. Solving the Îto equations
The general form of the Îto equation for a system of stochastic
differential equations in , where
is a stochastic process in *t*, is
where is a vector of functions, *G* is
a matrix of functions, is a vector of
independent Wiener processes and *D* is the correlation matrix of
the Wiener processes.
The moments of the elements of can be found
from the moment ordinary differential equation (ODE), see Soong
(1973), Sect. 7.1.4.1(c):
where .
### 3.1. Solving for moments in *µ* and *y*
The system of equations we wish to solve is (3), (4) and (5).
Fortunately, only one of the three equations contains an explicit
noise term, and further simplification arises because the energy
equation is very simple.
Firstly, we derive the statistical properties of
. This can be done by using the *µ*
Eq. (5) to construct the moment ODE using (7) with
which can be solved to express the *n* th moment as an
integral involving the th moment:
Next, using the same procedure, the moment ODE equation for
is
where can be determined from our final
moment equation
Together, (8), (9) and (10) can be used to find the moments of
. Doing so does not present any further
difficulties beyond the tedious task of evaluating multiple integrals
over *x*, where, in general, the integrands consist of terms of
the form , *l* being an integer.
To illustrate this, we calculate the first and second order
moments. Firstly, the moment equation for *µ* gives for
respectively:
After a little reduction, putting for
brevity, the first two moments of can be
expressed as
and
Result (13) is the familiar mean scattering result, re-derived from
this new rigorous standpoint. Result (14) is completely new, and gives
the variance in column depth of those electrons whose energy has been
degraded by a factor : the variance of y is
given by .
### 3.2. Constructing the distribution function
Now we show how the distribution function itself can be expressed
analytically. We note that such an exact, analytic expression is
mainly of academic interest, since the complete expression can be very
complicated, and may prove rather difficult to use in practise. More
often, the moments themselves will prove to be of more immediate use -
this is demonstrated in Sect. 5, where the second order moment is used
to look at how large pitch angle electrons spread away from their
point of injection.
The characteristic function of a random
variable *X* that has distribution function
is defined to be, see Soong (1973),
It is clear therefore that and *g* form
a Fourier transform pair. Now, the MacLaurin series for
is
Differentiating the integral in (15) gives
ie each term of the MacLaurin expansion of
is simply related to the moments of *X*.
Applying this to the random variable we find
that its distribution function is given by
where is the distribution *f*
integrated over pitch angle. The pitch angle distribution at *x*
can be calculated in a similar fashion and is known to be given by a
Legendre polynomial solution, see Kel'ner and Skrynnikov (1985), Lu
and Petrosian (1988) and MC.
© European Southern Observatory (ESO) 1998
Online publication: March 3, 1998
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