Diffusion in a tube of varying cross section: numerical study of reduction to effective one-dimensional description

Brownian dynamics simulations of the particle diffusing in a long conical tube (the length of the tube is much greater than its smallest radius) are used to study reduction of the three-dimensional diffusion in tubes of varying cross section to an effective one-dimensional description. The authors find that the one-dimensional description in the form of the Fick-Jacobs equation with a position-dependent diffusion coefficient, D(x), suggested by Zwanzig [J. Phys. Chem. 96, 3926 (1992)], with D(x) given by the Reguera-Rubi formula [Phys. Rev. E 64, 061106 (2001)], D(x)=D/sq rt1+R'(x)2, where D is the particle diffusion coefficient in the absence of constraints, and R(x) is the tube radius at x, is valid when |R'(x)|1, higher spatial derivatives of the one-dimensional concentration in the effective diffusion equation cannot be neglected anymore as was indicated by Kalinay and Percus [J. Chem. Phys. 122, 204701 (2005)]. Thus the reduction to the effective one-dimensional description is a useful tool only when |R'(x)|