  One-dimensional hopping model is useful to describe the motion of microscopic
particle in thermal noise environment, such as motor proteins. Recent
experiments about the new generation of light-driven rotary molecular motors
found that, the 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, such modified periodic
one-dimensional hopping model of arbitrary period N is studied mathematically.
The mean velocity, effective diffusion constant, and mean dwell time in one
single cycle are obtained. Corresponding results are illustrated and verified
by being applied to a type of synthetic rotary molecular motors.
