Diffusion Enhancement of Brownian Motors Revealed by a Solvable Model

A solvable model is proposed and analyzed to reveal the mechanism underlying the diffusion enhancement recently reported for a model of molecular motors and predicted to be observed in the biological rotary motor ${\mathrm{F}}_1$-ATPase. It turns out that the diffusion enhancement for the present model can approximately described by a random walk in which the waiting time for a step to occur is exponentially distributed and it takes nonzero time to proceed forward by the step. It is shown that the diffusion coefficient of such a random walk can significantly be increased when the average waiting time is comparable to the average stepping time.