Crossover from two- to one-dimensional Fickian diffusion in a quasi-one-dimensional system

In this work, we study the effects of geometric confinement on random walks and diffusion processes in systems of reduced dimensionality. Extensive Monte Carlo simulations of Gaussian random walks were performed on rectangular strips of infinite length. A special emphasis is made on the crossover from two- to one-dimensional diffusion in the Fickian regime. We found that the crossover behavior is controlled by the ratio of the strip width to the standard deviation of the walker step length distribution. Specifically, the characteristic time of crossover behavior scales quadratically with this ratio. Furthermore, the time dependence of the number of effective spatial degrees of freedom of the random walker on the strip is found to obey an ansatz characterized by the universal power-law exponent. This allows us to formulate the diffusion equation with the time dependent number of effective spatial degrees of freedom in the quasi-one-dimensional system.