  We develop non-equilibrium theory to account for non-ergodic effects that
arise due tomacroscopically "slow" fluctuations in heterogeneous systems. The
derived theory is broadly applicable to describe athermal fluctuations,
particularly those that do not obey detailed balance. An area of application
are fluctuations in capillary-dominated systems, which are non-stationary due
to the irreversibility of cooperative events. Considering droplet coalescence
alongside capillary fluctuations that occur for immiscible fluid displacement
in porous media, we show that a simple power law is insufficient to predict the
spectral power density $S_P$ for capillary fluctuations. Instead, we find that
$S_P$ is linked to frequency $F$ based on an exponential relationship, $\log
S_P \sim \alpha F^p$, with $\alpha=-15$ and $p=1/2$ describing the fluctuations
of both systems.
