  Thermally activated particle motion in disorder potentials is controlled by
the large-$\Delta V$ tail of the distribution of height $\Delta V$ of the
potential barriers created by the disorder. We employ the optimal fluctuation
method to evaluate this tail for correlated quenched Gaussian potentials in one
dimension in the presence of a small bias of the potential. We focus on the
mean escape time (MET) of overdamped particles averaged over the disorder. We
show that the bias leads to a strong (exponential) reduction of the MET in the
direction along the bias. The reduction depends both on the bias, and on
detailed properties of the covariance of the disorder, such as its derivatives
and asymptotic behavior at large distances. We verify our theoretical
predictions, as well as earlier predictions for zero bias, by performing
large-deviation simulations of the potential disorder. The simulations employ
correlated random potential sampling based on the circulant embedding method
and the Wang-Landau algorithm, which enable us to probe probability densities
smaller than $10^{-1200}$.
