What is a real model of thermal electrical noise? A common model for electrical noise is to assume a flat frequency distribution, i.e. white noise. However, this results in theoretically infinite energy. So my question is: What has to be accounted for when going from ideal electrical white noise to more realistic noise? Is it the RLC-properties/transfer function of the circuit?
 A: In his original analysis Nyquist [1] treated the "low" frequency, ie., RF case in which the power spectral density is flat $S(f) = k_B T$ and the noise emf in a resistor is $\langle e^2(f)df\rangle  = 4R(f) k_BT df$.
Nyquist also mentions that the full quantum formula for the power spectral density is
$$S(f) = \frac {hf}{e^{hf / k_B T}-1}$$
with the corresponding mean square emf be
$$\langle e^2(f) df \rangle= 4R(f) \frac {hdf}{e^{hf/k_B T}-1} \, .$$
As long as $hf \ll k_BT$ the quantum effects are negligible and we can Taylor expand the exponent in the Nyquist formula to get
$$
S(f) \approx \frac{hf}{1 + (hf / k_B T) - 1} = k_B T
$$
which doesn't depend on frequency, i.e. it's flat.
This flat formula, called the Johnson formula, is used daily to design/operate every single RF comm/radar system in existence.
At what range of frequency and temperature is the Johnson formula valid?
Putting in numbers for $k_B$ and $h$ we find the condition $hf \ll k_B T$ can be written as
$$
\frac{f / 1 \, \text{GHz}}{T / 1 \, \text{Kelvin}} \ll 20 \, .
$$
So for example, at a temperature of 1 Kelvin, the Johnson formula breaks down when the frequency is at or above 20 GHz.
At room temperature (300 Kelvin) the Johnson formula breaks down at around 6 THz.
So as we can see, except at very low cryogenic temperatures or high frequencies the white noise Johnson formula is good. In the optical regime the situation is, of course, different.
Note that the meaning of a lumped element resistor/capacitor/inductor is questionable at frequencies where the wavelength is not much longer than the size of the R/L/C element.
[1] Nyquist: "THERMAL AGITATION OF ELECTRIC CHARGE IN CONDUCTORS", Phys. Rev. July 1928
