Moment expansion for mapping of the confined diffusion

The mapping of diffusion in a 2D channel with varying cross section A(x) onto the longitudinal coordinate x is revisited. We present an algorithm based on construction of a specific hierarchy of equations for the transverse moments of the 2D density. Elimination of all the moments but the zeroth one, the 1D density p(x, t), results in the mapped equation. Our calculation validates the earlier mapping procedure [P. Kalinay and J. K. Percus, Phys. Rev. E 74, 041203 (2006); 78, 021103 (2008)], presuming existence of the backward mapping operator, and it naturally arrives at the extended Fick-Jacobs equation [D. Reguera and J. M. Rub` i, Phys. Rev. E 64, 061106 (2001)] in the stationary flow, without any phenomenological conjectures.