Biharmonic Fick–Jacobs diffusion in narrow channels

We study the effect of a fourth-order derivative term in the diffusion of Brownian particles confined to a narrow 2D channel whose longitudinal coordinate is larger than the transversal one. Fourth-order terms arise in some situations, such as at high densities with long-range effects, diffusivity switching process approximations, or aggregation models. In these cases, the diffusive flux does not adequately describe the system’s behavior, so the biharmonic terms help model slight deviations from standard Fick behavior. Here, a Fick–Jacobs-like equation is found using the projection method. It contains a third-order, in addition to the standard first-order entropic flux. It is shown that even at the lowest order, position-dependent modifications to the longitudinal diffusivity and the drift term appear, which also depend on a new scale related to the coefficient of the fourth-order term. The formal stationary solution of such equation is also discussed. Furthermore, using a simplified Kalinay–Percus method, the transport coefficients in the first, second, and third derivatives, all functions of the longitudinal coordinate and with terms at the new scale, are obtained. Straight-walled channels are analyzed to illustrate the results. Surprisingly, although small, there are non-negligible effects due to the new scale. For instance, there is a region where diffusion can be enhanced, and the induced entropic drift reversed.