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-Rubí formula [ Phys. Rev. E 64 , 061106 ( 2001 )] , D ( x ) = D ∕ 1 + 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 . 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 ) ∣ ⩽ 1 since in this case one can apply the powerful standard methods to analyze the resulting diffusion equation.