Optimal transport and phase transition in dichotomic ratchets

We revisit the problem of transport of an overdamped particle in spatially periodic potentials which is driven by exponentially correlated dichotomic noise of zero mean value. There are three transport regimes: transportless, non-diffusive and diffusive. The stationary average velocity of the particle can exhibit a behavior similar to a second-order phase transition with respect to an ‘order parameter’ being the noise amplitude: for a small noise amplitude it is zero. When the noise exceeds the critical amplitude, the non-diffusive regime occurs in which the velocity is an increasing (or decreasing) function of the noise amplitude. Near the onset of the transition point, it obeys the power-law scaling with the exponent which depends on order of the force at its global minimum. In the diffusive regime, the stationary average velocity is a decreasing (or increasing) function of the noise amplitude. The maximal absolute velocity is for the noise amplitude which separates the non-diffusive and diffusive regimes.