We present numerical results for the transport and diffusion of underdamped Brownian particles in one-dimensional disordered potentials. We compare the anomalies observed with those found in the overdamped regime and with results for a periodic potential. We relate these anomalies to the time dependent probability distributions for the position and velocity of the particles. The anomalies are caused by the random character of the barrier crossing events between locked and running states which is manifested in the spatial distributions. The role of the velocities is small because the particles quickly thermalize into locked or running states.
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