I am trying to understand how noise in electrical circuits can be modeled via stochastic differential equations (SDE). For example, the current through a resistor can be described as the sum of a noiseless, deterministic current and a current due to a Gaussian white noise process with spectral density $S(f)=\frac{2k_BT}{R}$, where $K_B$ is the Boltzmann constant, $T$ the temperature and $R$ the resistance of the resistor.
The additional current is then modelled as \begin{equation} I(t)=\sqrt{\frac{2k_BT}{R}}\xi(t), \end{equation} where $\xi(t)$ is a white noise process. This would enter the SDE as $\sqrt{\frac{2k_BT}{R}}dW(t)$, where $W(t)$ is a Wiener process.
Why do you take the square root of the spectral density here? Why does this white noise process have spectral density $\frac{2k_BT}{R}$? I know that the spectral density of a stochastic process is the Fourier transform of its auto-correlation function. But it seems I am missing something here.
The square root is a normalization. Note that $$\mathbb{E}(\xi(t) \xi(t + s)) = \delta(s)$$ and the Fourier transform of $\delta$ is just the constant $1$ everywhere. Thus, if you want white noise with spectral density $C$ instead of $1$, you use $\xi_{C} = \sqrt{C} \xi$ instead of $\xi$, which gives you $$\mathbb{E}(\xi_{C}(t) \xi_{C}(t + s)) = C \delta(s)$$ which has spectral density $C$.
A physicist might say you want the energy dissipation of the "random current" to be $C = \frac{k_{B} T}{R}$ so the random current should be $\sqrt{C} \xi$. After all, energy dissipates like current squared.
The constant $\frac{k_{B}T}{R}$ is from physical considerations. I remember seeing it in EE classes, but that's as much as I can say. I have a hunch there's a derivation in Kittel's Thermal Physics; if you like, I could report back when I get home. At any rate, Boltzmann's constant is something fundamental about statistical physics, often it's multiplied by $T$, and, well, you know what $R$ is.
Edit: I didn't find $\frac{k_{B}T}{R}$ in Kittel's book. I'll try EE textbooks at some point and get back to you.