*Astron. Astrophys. 331, 1103-1107 (1998)*
## 2. Fokker-Planck equation
The Fokker-Planck equation for non-relativistic electrons in a cold
hydrogen plasma is
where , *t* is time measured from some
specified initial condition, *z* is the physical distance
measured *along* the field line, *v* is speed,
is the mass of the electron, *µ* is
the electron's pitch angle and the units of *f* are (cm
s^{-1})^{-1} cm^{-3}. The initial distribution
at is taken to be
and as described in MC, linear superposition of solutions with
different initial conditions can yield the solution at time *t*
for any arbitrary initial distribution function, or with some added
source term on the right hand side.
In terms of Îto calculus, the Fokker-Planck equation may be
expressed as a set of three stochastic differential equations:
where , ,
at and *dW* is a
Wiener process (see below). Upon a change of variables to
:
where represents a Wiener process which has
mean zero, variance 2 and auto-covariance given by
, where is the Dirac
delta function. A Wiener process can be thought of as a continuous
white noise (time) series in *x* that is independently
distributed at each *x*. The variable *x* is the normalised
column depth "seen" by the electron, ie the column depth integrated
along the electron's path. Intuitively, *x* can be thought of as
being the electron's "life-time" parameter as it depends solely on the
fraction of the electron's energy . The variable
*y* is the normalised column depth integrated along the
electron's motion projected onto the field line: *y* can be
identified with the column depth *N* appearing in mean scattering
theory (1). The normalisation is by the stopping column depth measured
along the electrons path, given by (2).
The aim is to derive the statistical moments
, where *n* is a positive integer, and
where is the expectation operator. These
moments, in physical terms, will provide information on the particles'
spatial distribution along the field when the particles have been
degraded to energy .
At this point, it is clear from the form of (3), (4) and (5)
that
so that for a given *x* the moments only depend on
, and not explicitly on the injection energy
. It can easily be seen that the *n* th
moment of the column depth itself is which
scales as . From this we can state quite
generally that the mean, variance, skewness, kurtosis and so on of the
electron's column depth measured along the field, at the point in its
life given by *x*, are all simply proportional to
, ,
, and so on,
respectively. For example, if many electrons were injected with energy
, then their mean position at the end of their
lives would be proportional to (as predicted by
mean scattering) and their spread about that position (ie the root
variance) would also be proportional to . The
following section shows how to derive the functions
.
© European Southern Observatory (ESO) 1998
Online publication: March 3, 1998
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