What could be and how to come up with a mathematical formula for the local electric field inside a blackbody cavity?

Question

I asked a similar question here more than two years ago. I did not get an answer to my complete satisfaction. I would like to reiterate the problem again.
The local electric field of a monochromatic radiation is nonzero and varies sinusoidally in a predictable fashion. For example, the electric field of an ideal monochromatic radiation is described by $${bf E}({bf r},t)={E}_0hat{{ varepsilon}}cos({bf k}cdot{bf r}-omega t),$$ at any location ${bf r}$, is nonzero at any time $t$ and varies with time in a predictable manner. Here, $E_0$ is a fixed number and so is $omega$ (the frequency of the radiation), and $hat{varepsilon}$ represents the constant polarization vector.
In contrast, assuming that the electric field at any location of a Blackbody cavity is due to an incoherent superposition of electric fields of all frequencies, polarizations (and all amplitudes?), can we rigorously come up with a mathematical expression for the local electric field at any time $t$ for the blackbody radiation? I am interested in getting a mathematical formula that properly represents the local electric field of incoherent thermal radiation.

ProfRob · Answer

Using Cartesian coordinates you could write something like:
begin{eqnarray} 
vec{E}(vec{r},t) &  =  & int E_{0}(omega) left(cos (omega[t - x/c]) hat{j} + cos(omega[t -x/c] +phi_{x,omega}) hat{k}right) domega nonumber 
           & + & int E_0(omega)left( cos (omega[t - y/c]) hat{i} +  cos(omega[t -y/c] +phi_{y,omega}) hat{k}right) domega nonumber 
           & + & int E_0(omega)left( cos (omega[t - z/c]) hat{i} +  cos(omega[t -z/c] +phi_{z, omega}) hat{j}right) domega . nonumber 
end{eqnarray}
This represents the sum of three unpolarised beams of light travelling in three directions, each with equal power, summed over all frequencies. In each term, the $phi_{i,omega}$ term represents a random phase between $0$ and $2pi$.
The time averaged value of $vec{E}cdot vec{E}$ is
$$left< vec{E}cdot vec{E}right> = 3 int E_0^2(omega) domega    $$
and so the time-averaged energy density of the electromagnetic fields would be
$$ 6epsilon_0 int E_0^2(omega) domega = int frac{hbar omega^3}{pi^2 c^4} left(frac{1}{exp[hbar omega/k_BT]-1}right) domega , $$
where the term on the right is the total energy density of a blackbody radiation field and the extra factor of 2 on the left hand side accounts for equal energy density in the magnetic field.
Thus
$$E_0(omega) = left[ frac{hbar omega^3}{6epsilon_0pi^2 c^4} left(frac{1}{exp[hbar omega/k_BT]-1}right) right]^{1/2} $$
Of course, the time-average of the electric field itself is zero.

What could be and how to come up with a mathematical formula for the local electric field inside a blackbody cavity?

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