Mapping of diffusion in a channel with soft walls

We study diffusion of pointlike particles biased toward the x axis by a quadratic potential U(x,y)=κ(x)y2. This system mimics a channel with soft walls of some varying (effective) cross section A(x), depending on the stiffness κ(x). We show that diffusion in this geometry can also be mapped rigorously onto the longitudinal coordinate x by a procedure known for channels with hard walls; i.e., we arrive at a one-dimensional evolution equation of the Fick-Jacobs type. On the other hand, the calculation presented serves as a prototype for mapping of the Smoluchowski equation with a wide class of potentials U(x,y) varying in both the longitudinal as well as the transverse directions, which is necessary for understanding, e.g., stochastic resonance.