A simple example that highlights the strength of this method: a 137Cs point source is at the origin. A detector consisting of a right cylinder at arbitrary distance and orientation is nearby. What is the solid angle?
There may exist an analytical solution for this, but I wouldn't trust myself to derive it correctly. It would certainly be a huge mess.
If we add that the source is also a right cylinder instead of point source, and we want to add first order attenuation of emitted gammas by the source itself, the spreadsheet becomes only a bit more complex, but there will not be a pen and paper equation solution.
In this example every row of the spreadsheet would represent a hypothetical ray. One could randomly choose a location in the source, a random trajectory, and check if the photon intersects the detector. An alternative approach would be randomly choosing points in both target and detector, then doing additional math.
The results are recovered by making histograms and computing stats on the outputs of all the rows. You probably need a few thousand for most things at least. Remember roughly speaking 10k hits gets you ~1% statistics.
What annoys me with Monte Carlo methods is trying to get self-consistent statistics at multiple parameter values. E.g., in your example, what is the derivative of the solid angle as the detector is moved? Or more generally, if we have some 'net goodness' measure for a system depending on parameters, how can we efficiently maximize it, when simulations are noisy and basins are shallow?
My understanding is that these sorts of questions come up in ML, and there are ways of dealing with it, but they can't converge nearly as fast as simple iterations like Newton's method. Even if I have to take a series approximation instead of a simple formula, I'll be able to use autodiff (or at worst, symbolic differentiation) to get quick and precise answers to these questions.
There may exist an analytical solution for this, but I wouldn't trust myself to derive it correctly. It would certainly be a huge mess.
If we add that the source is also a right cylinder instead of point source, and we want to add first order attenuation of emitted gammas by the source itself, the spreadsheet becomes only a bit more complex, but there will not be a pen and paper equation solution.
In this example every row of the spreadsheet would represent a hypothetical ray. One could randomly choose a location in the source, a random trajectory, and check if the photon intersects the detector. An alternative approach would be randomly choosing points in both target and detector, then doing additional math.
The results are recovered by making histograms and computing stats on the outputs of all the rows. You probably need a few thousand for most things at least. Remember roughly speaking 10k hits gets you ~1% statistics.