  We introduce an explicit solution for the non-equilibrium steady state (NESS)
of a ring that is coupled to a thermal bath, and is driven by an external hot
source with log-wide distribution of couplings. Having time scales that stretch
over several decades is similar to glassy systems. Consequently there is a wide
range of driving intensities where the NESS is like that of a random walker in
a biased Brownian landscape. We investigate the resulting statistics of the
induced current $I$. For a single ring we discuss how $sign(I)$ fluctuates as
the intensity of the driving is increased, while for an ensemble of rings we
highlight the fingerprints of Sinai physics on the $abs(I)$ distribution.
