Mapping of diffusion in a channel with abrupt change of diameter

Mapping of the diffusion equation in a channel of varying cross section onto the longitudinal coordinate is already a well studied procedure for a slowly changing radius. We examine here the mapping of diffusion in a channel with abrupt change of diameter. In two dimensions, our considerations are based on solution of the exactly solvable geometry with abruptly doubled width at $x=0$. We verify the surmise of Berezhkovskii et al. [J. Chem. Phys. 131, 224110 (2009)] that one-dimensional diffusion behaves as free in such channels everywhere except at the point of change, which looks like a local trap for the particles. Applying the method of “sewing” of solutions, we show that this picture is valid also for three-dimensional symmetric channels.

Figure 1

Conformal transformation [Eq. (14)] transforms the upper half plane (the upper figure) onto the stepwise channel (the lower figure).

Figure 2

Contour plot of the 2D density $\rho \left(x,y\right)$ in a channel of widths $a=\pi /2$ and $b=\pi$ for $x>0$ and $x<0$, respectively.

Figure 3

Shape of the 2D channel.

Figure 4

Shape of the 3D channel.