I've been doing a lot of reading on Johnson-Nyquist (thermal) noise lately, and sadly much of the material I've come across either looks at things from a very engineering prospective (that is, little theory/explanation/derivation, very formulaic and applied) or is highly technical and, admittedly, way over my head.
I've tried looking at a couple derivations (e.g http://123.physics.ucdavis.edu/week_2_files/Johnson_noise_intro.pdf) based on the original Johnson and Nyquist papers, but everything I find offers a minimum of explanation and/or is rather esoteric. I have what I think is a good high-level intuition for the phenomena, and I know all the formulas and how to apply them, but I'd like a deeper understanding.
How were the formulas derived (e.g., the power spectral density function $P = 4kTR$, etc)? Why is it, on a very low level, that the thermal noise translates to power/voltage (i.e. what is the low-level mechanism behind this?).