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 $F\left(t\right)=F_{0}cos\omega t$. We arrive at analytic formulas for the leading term of the ratchet current $J_{r,2}\left(\omega \right)\sim F_0^2$ for any corrugation of the channel $h\left(x\right)$.

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• Brownian motion
• Diffusion
• Nonequilibrium statistical mechanics
• Brownian ratchet
• Fokker–Planck equation
• Linear response theory