Mapping of diffusion in a channel with soft walls

摘要: We study diffusion of pointlike particles biased toward the $x$ axis by a quadratic potential $U(x,y)=\ensuremath{\kappa}(x){y}^{2}$. This system mimics channel with soft walls some varying (effective) cross section $A(x)$, depending on stiffness $\ensuremath{\kappa}(x)$. show that in this geometry can also be mapped rigorously onto longitudinal coordinate procedure known for channels hard [P. Kalinay and J. K. Percus, Phys. Rev. E 74, 041203 (2006)]; i.e., we arrive at one-dimensional evolution equation Fick-Jacobs type. On other hand, calculation presented serves as prototype mapping Smoluchowski wide class potentials $U(x,y)$ both well transverse directions, which is necessary understanding, e.g., stochastic resonance.