Mapping of diffusion in a channel with soft walls

We study diffusion of pointlike particles biased toward the $x$ axis by a quadratic potential $U(x,y)=\kappa (x)y^2$. This system mimics a channel with soft walls of some varying (effective) cross section $A(x)$, depending on the stiffness $\kappa (x)$. 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 [P. Kalinay and J. K. Percus, Phys. Rev. E 74, 041203 (2006)]; 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.

Figure 1

Potential $U(x,y)=\kappa (x)y^2$ for the stiffness $\kappa (x)=1+(1/2)\cos 2\pi x$.

Figure 2

Plot of the effective diffusion coefficient $D(x)$ versus derivative of the width (or the effective width) $A^{\prime }(x)$ of the channel in the linear approximation. The thick lines represent the formulas for the channels with hard walls; the solid line depicts the exact sum (1.4) for $\epsilon =1$ and $D_0=1$; the dashed line corresponds to the approximation of Reguera and Rubí [3], $D(x)=(1+A^{\prime 2}){}^{-1/3}$. The thin lines describe $D(x)$ for the channels with soft walls, defined by the potential $U(x,y)=\kappa (x)y^2$, for various scaling of the effective width of the channel $A(x)$. The dashed line depicts formula (2.28), taking $A(x)=\sqrt{\pi /\alpha (x)}$, $\alpha (x)=\kappa (x)/k_{B}T$. The dotted line shows the same formula with $A(x)$ rescaled to fit $A^2(x)=1/2\alpha (x)$ according to Eq. (2.4). The solid line represents the result for $A(x)=\sqrt{3/2\alpha (x)}$, which fits $D(x)$ for the channels with hard walls at small $A^{\prime }$.