Test Nanbu implementation against basic test problems from paper

[MatthewGrim]

This issue tests the implementation of Nanbu against a set of simulations from the paper:

Theory of cumulative small-angle collisions in plasmas - K. Nanbu

The aim of the tests if to validate the simple version of Nanbu, which only uses two species.

[MatthewGrim]

The single species relaxation shows an asymmetry in profiles, where the x velocity profile has a larger spread than the other two. This is independent of the random number generator seed.

[MatthewGrim]

The likely culprit is the h vector specified in the momentum change calculation. As the terms in z and y velocity are similar, but differ for x... I don't know what I am missing here.

[MatthewGrim]

This equation is the same as that used in Takizuka and Abe though... just re-formulated

[MatthewGrim]

An alternative way of viewing the issue is that the scattering angle chi being generated for the cumulative scatter has a problem - this would be in line with the issues that I am seeing in the electron thermalisation simulations.

[MatthewGrim]

Issue is invariant to changes in the average velocities of the particles ... my guess was wrong

[MatthewGrim]

In the electron thermal relaxation, the time scale is probably wrong because the coulomb logarithm in the theory is fixed. We need to add this option into the simulation.

[MatthewGrim]

The results are wrong in the electron thermal relaxation for the same reasons as the single species case. There is something wrong with the collision that makes the x and y/z distributions different.

[MatthewGrim]

I've tested the sample function from the paper and it is definitely an accurate sample of the PDF, as shown below:

[MatthewGrim]

The solutions all seem much more reasonable - embarrassing it all came down to a missing minus sign...