Background
I recently asked this question over on Electrical Engineering Stack Exchange. On the advice of some commenters there, I've broken off those pieces which are appropriate for asking as smaller questions here. Namely, I'm asking separately about modeling the signal chain, and a more accurate theoretical model of the behavior of noise itself, and the resulting antenna response.
I'm attempting to write a physics simulation code, one portion of which involves the simulation of the voltage observed at a set of \$N\$ radio antennae immersed in some medium due to thermal (or "Johnson-Nyquist") noise, and which may expand to include other noise sources (e.g. triboelectric, anthropogenic) in the future.
For now, I do so by modeling thermal noise simplistically as Gaussian-distributed white noise centered on \$0\$ and with \$V_{RMS} = \sqrt{4k_BTBR}\$. The voltage "waveforms" are produced by drawing \$N\$ sets of \$\textrm{sampling rate} \times \textrm{duration}\$ samples from the Gaussian distribution.
After some discussion with faculty advisors, I've decided that I'd like to improve upon this simplistic model (largely because we find it insufficiently close to data). I'd like to begin by developing a more sophisticated theoretical model of the behavior of thermal noise, and how it is observed at the antenna, and that's what I'll discuss here.
Where My Understanding Lies
Much of this is admittedly motivated by my lack of a fully confident understanding of what, precisely, is actually measured by an antenna. Here's my understanding of the problem thus far:
In most texts, the Gaussian model is developed by approximating the electric field vectors due to thermal noise as complex phasors whose real and imaginary components are identically and independently Gaussian distributed (this coming, obviously, from the central limiting behavior of the contributions of a large number of e.g. atoms or molecules). When the phasor is observed at the antenna, the real part is what is observed and digitized (corresponding to the in-phase part of the electric field magnitude). The magnitude of the real part is distributed according to a folded Gaussian, while the magnitude of the phasor itself is Rayleigh distributed.
The actual electric field vectors, however, are \$3\$-dimensional, of course, and their magnitudes are distributed according to a "\$3\$D" Rayleigh distribution (this likely has a more precise name, however I'm unable to find one). Below, I've plotted the distribution of magnitude of the real part of a large number number of complex phasors (in blue), the distribution of the "full" magnitude of these phasors (in pink), and the distribution of the magnitude of a large number of \$3\$D vectors (in red):
I had thought that what is measured at the antenna was the magnitude of this \$3D\$ electric field vector, which follows a very different distribution from the real part of the complex phasor.
What I'd Like to Achieve
What is measured at the antenna, and how is it distributed? That is, when I look at the voltage measured due to thermal noise (or more generally), what can I expect to see? Is it the magnitude of the electric field vector, or the real part of the complex phasor? Why?
How is this effected by the properties of the antenna? Suppose I know the dimensions, material composition, polarization, response (as a function of frequency), and radiation pattern of the antenna. How does this effect the voltage that is observed due to thermal noise, if at all? I know for one that, in the simple Gaussian model, an ideal rectangular passband is assumed (hence, in a more sophisticated model, I might left $B$ be the integral under the true passband (modeled e.g. by a Butterworth filter)).
I understand that this is likely a rather involved task, and that what is sufficiently complex in any physics model is subjective. I'm looking for (ideally mathematically-motivated) suggestions for how to move forward, and resources for further reading (ideally accessible to a (perhaps slightly advanced) upper-level undergraduate). As of now, I simply don't know where to begin (nor what would be a good framework to begin with that would allow for additional layers of complexity to be easily added on).
I'd also simply like to achieve a deeper theoretical understanding. I'm open (and encourage) detailed and complex theoretical answers if necessary. I can trim those down to something I can apply for my practical use case.