Periodic one-dimensional hopping model with transitions between nonadjacent states

A one-dimensional hopping model is useful for describing the motion of microscopic particles in a thermal noise environment. Recent experiments on the new generation of light-driven rotary molecular motors found that a motor in state i can jump forward to state i + 1 or i + 2 or backward to state i - 1 or i - 2 directly. In this paper, inspired by these experiments, such a modified periodic one-dimensional hopping model with arbitrary period N is studied theoretically. The mean velocity, effective diffusion constant, and mean dwell time in one single mechanochemical cycle are obtained. The corresponding results are illustrated and verified by being applied to the synthetic rotary molecular motors.