Diffusion in zeolites, biological membranes, or porous media is usually confined to narrow channels, which the diffusion process is confined to single file (mutual passage of particles in channels is prohibited). Non-Fickian behavior of single file diffusion was found for particles in stochastic background.

In the thesis, we will first present a mathematical procedure for obtaining the probability distribution function (PDF) of a single file diffusion (SFD) system. Given the isolated particle dynamics, the PDF of SFD is simply a closed form formula. The time dependence of the mean square displacement (MSD) is also studied and compared to experimental results.

We also formulate the SFD on a simple lattice chain. The criteria for the equivalency between continuous and lattice system will be presented. In general, diffusion on a lattice is found to share the same time dependence with diffusion in a continuum media. The effect of the presence of multiple occupancies in lattice system is also studied.

For theoretical completeness, we will then extend our density functional formalism from the real space to the phase space along the line of Lebowitz and Percus' paper. In particular, for inertial particle dynamics, the kinetic equations will be written down.

Finally we will extend our one dimensional system to quasi-one dimension (q1D), and will refine the approximation of the effective particle size in q1D. We will make comparison with simulations of hard disc in a channel.