Rocking ratchet revisited via the Stratonovich formula

Diffusion in the channels with flexible moving boundaries as well as driven by time-dependent forces can be described by the generalized Fick-Jacobs (Smoluchowski) equation in the reduced 1D picture, with the Boltzmann factor A(x,t) and the effective diffusion coefficient D(x,t) depending not only on the longitudinal coordinate x, but also on time t. We show here that for the driving periodic in time and space, the asymptotic solution of the rocking density and the rectified current can be found consistently as a series of corrections to the adiabatic limit given by the Stratonovich formula. The solution is tested on the simplest rocking ratchet system, driven by the homogeneous sinusoidal force F(t)=F_{0}cosωt. We arrive at analytic formulas for the leading term of the ratchet current J_{r,2}(ω)∼F_{0}^{2} for any corrugation of the channel h(x).