A "Burnt Bridge'' Brownian Ratchet
  Motivated by a biased diffusion of molecular motors with the bias dependent on the state of the substrate, we investigate a random walk on a one-dimensional lattice that contains weak links (called "bridges'') which are affected by the walker. Namely, a bridge is destroyed with probability p when the walker crosses it; the walker is not allowed to cross it again and this leads to a directed motion. The velocity of the walker is determined analytically for equidistant bridges. The special case of p=1 is more tractable -- both the velocity and the diffusion constant are calculated for uncorrelated locations of bridges, including periodic and random distributions. 
T. Antal and P. L. Krapivsky