  Ion transport through biological and solid-state nanochannels is known to be
a highly noisy process. The power spectrum of current fluctuations is
empirically known to scale like the inverse of frequency, following the
long-standing yet poorly understood Hooge's law. Here, we report measurements
of current fluctuations across nanometer-scale two-dimensional channels with
different surface properties. The structure of fluctuations is found to depend
on channel's material. While in pristine channels current fluctuations scale
like $1/f^{1+a}$ with $a = 0 - 0.5$, the noise power spectrum of activated
graphite channels displays different regimes depending on frequency. Based on
these observations, we develop a theoretical formalism directly linking ion
dynamics and current fluctuations. We predict that the noise power spectrum
take the form $1/f \times S_\text{channel}(f)$, where $1/f$ fluctuations emerge
in fluidic reservoirs on both sides of the channel and $S_\text{channel}$
describes fluctuations inside it. Deviations to Hooge's law thus allow direct
access to the ion transport dynamics of the channel -- explaining the entire
phenomenology observed in experiments on 2D nanochannels. Our results
demonstrate how current fluctuations can be used to characterize nanoscale ion
dynamics.
