Re: [ESPResSo] bug in random.tcl in mbtools

[Markus Deserno]

Hi,

sorry for repeating Torstens email, just missed it!
However, one more comment on this:

> [...] It is based on the (not very intuitive) fact, that
> the projection of homogeneously distributed points on a
> sphere-surface onto any axis gives a normal distribution
> of points on that axis (a prove of that can be found
> in math books).

That's not true if you read "normal" in the normal sense
of "Gaussian".  The projections are really equally distributed
in the sense of a flat distribution between -R and R.

But contrary to Torsten's despair this is in fact very
intuitive, if you recall a bit of calculus:  When you
integrate over the sphere, you do not just use the area
element d(theta) d(phi).  We all know that due to
"coordinate pile up" at the poles these regions would
be over-weighted, and that the proper way to correct
for this is to include the Jacobi functional determinant
|sin(theta)|.  However:

|sin(theta)| d(theta) d(phi) = |d(cos(theta))| d(phi)

And since cos(theta) is just the projection on the z-axis,
let's now just call it z, you realize that you achieve an
equal weighting of every region on the sphere if you measure
it in z-phi-coordinates.  And these are exactly the coordinates
which are drawn equidistributed in the algorithm that Torsten
proposed.

Hope that demystifies things a bit.

Best,

Markus

-- 
Dr. Markus Deserno
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