ac - RMS current of a circuit driven by a white noise voltage source

Note: I am an applied mathematician, not an electrical engineer, so apologies if this comes across as silly. I am analyzing the following circuit as part of some modeling related to energy storage systems:

^{simulate this circuit – Schematic created using CircuitLab}

Where the voltage source is gaussian white noise (hence my use of "mixed Hz" in the schematic software this site uses). This means that the fourier transform of the voltage waveform is flat across all frequencies.

I was wondering if, given the rms Voltage, I could calculate an "equivalent capacitative reactance" \$X_{c,eq}\$ such that:

$$I_{rms} = \frac{V_{rms}}{X_{c,eq}}$$

I've heard of Johnson-Nyquist noise and kTC noise for capacitors, but I don't see how temperature applies here (since they are modelling a physical phenomena).