# Modeling stochastic process with frequency-dependent power spectrum

I'm trying to model of Johnson-Nyquist noise propagation in a nonlinear circuit. An ideal (linear) resistor can be modeled very nicely by the Fokker-Planck equation (equivalently, the drift-diffusion equation), where charge $V/R$ flows through the resistor on average, but there's also random flow of charge either way across the resistor characterized by "diffusion coefficient" $k_BT/R$. I get a nice differential equation describing how charge probabilistically flows through my circuit. Everything is good.

Then stage 2 is to have a resistor with frequency-dependent resistance (like all resistors in the real world). Here I get stuck...

A time-domain-based analytical solution seems impossible because---with frequency-dependent resistance---it would seem that charge transport across the resistor right now depends on the entire history of charge transport in the past.

A time-domain numerical (monte carlo) solution seems impossible because the relevant frequencies vary over many orders of magnitude I don't know how to construct a time-domain stochastic model with a predetermined power spectrum.

Any kind of frequency-domain solution seems impossible because other parts of the circuit are extremely nonlinear and therefore mix different frequencies together.

Any advice? Am I missing some trick?

• I'm out of my depth here, but something like a Drude model (ref wikipedia) for the resistor could be implemented by adding a small series inductance. – Art Brown Jun 13 '12 at 17:15