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Astron. Astrophys. 331, 1103-1107 (1998)

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3. Solving the Îto equations

The general form of the Îto equation for a system of stochastic differential equations in [FORMULA], where [FORMULA] is a stochastic process in t, is

[EQUATION]

where [FORMULA] is a vector of functions, G is a matrix of functions, [FORMULA] is a vector of independent Wiener processes and D is the correlation matrix of the Wiener processes.

The moments of the elements of [FORMULA] can be found from the moment ordinary differential equation (ODE), see Soong (1973), Sect. 7.1.4.1(c):

[EQUATION]

where [FORMULA].

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 [FORMULA]. This can be done by using the µ Eq. (5) to construct the moment ODE using (7) with [FORMULA]

[EQUATION]

which can be solved to express the n th moment as an integral involving the [FORMULA] th moment:

[EQUATION]

Next, using the same procedure, the moment ODE equation for [FORMULA] is

[EQUATION]

where [FORMULA] can be determined from our final moment equation

[EQUATION]

Together, (8), (9) and (10) can be used to find the moments of [FORMULA]. 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 [FORMULA], l being an integer.

To illustrate this, we calculate the first and second order moments. Firstly, the moment equation for µ gives for [FORMULA] respectively:

[EQUATION]

[EQUATION]

After a little reduction, putting [FORMULA] for brevity, the first two moments of [FORMULA] can be expressed as

[EQUATION]

and

[EQUATION]

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 [FORMULA]: the variance of y is given by [FORMULA].

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 [FORMULA] of a random variable X that has distribution function [FORMULA] is defined to be, see Soong (1973),

[EQUATION]

It is clear therefore that [FORMULA] and g form a Fourier transform pair. Now, the MacLaurin series for [FORMULA] is

[EQUATION]

Differentiating the integral in (15) gives

[EQUATION]

ie each term of the MacLaurin expansion of [FORMULA] is simply related to the moments of X.

Applying this to the random variable [FORMULA] we find that its distribution function [FORMULA] is given by

[EQUATION]

where [FORMULA] 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.

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© European Southern Observatory (ESO) 1998

Online publication: March 3, 1998
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