Alright so I am still not 100% on the physics of all of this, but my issues are fairly minor conceptual issues (stuff like "why does it scatter in a speherical wave?" or "how exactly do we know that the electric field is described in this way?"). I think I understand enough to actually move forward with the simulation. So we start with \(\phi_j = \textbf{K} \cdot \textbf{r}_j\). In this, \(\phi_j \) is the phase of the scattered electric field, \( \textbf{r}_j \) is the vector describing a "step" of the brownian motion, and \( \textbf{K} \) is the scattering vector which is equivalent to the difference between the initial and final wave vectors.
With the phase we can find the electric field \(E_0 e^{i(\phi - \omega_0 t)}\). Of course that is just the electric field from a single particle scattering, the electric filed at the detector would be the sum \(E(t) = \sum_{j = 1}^{N} E_0 e^{i(\phi_j(t) - \omega_0 t)} \) where N is of course all illuminated spheres. From this we can calculate the intensity at the the detector from \(I(t) = \beta E_0^2 \sum_{j}^{} \sum_{k}^{} e^{i(\phi_j(t) - \phi_k(t))}\)
After we get the intensity we calculate the auto correlation function and power series. Technically those are calculated with the voltage not the intensity, but except for some low background noise the intensity is directly proportional to the voltage so any analysis of the power series and auto correlation function would be true for both voltage and intensity. I still need to figure out what exactly that analysis should be or what power series and auto correlation functions are in general and also more to the math behind them. Disregarding that part for now, this is hopefully enough to start with.
