2. Fokker-Planck equation
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 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