A kinetic theory is presented for the transport (collective) diffusion of molecules residing in a one dimensional periodic potential and relaxing to a bath formed by the substrate. Observing that the behavior of the system on a macroscopic scale cannot depend on the detailed rapidly varying motion of the molecules on the microscopic scale of the potential, and assuming that the molecules bound in the well can contribute only indirectly to the macroscopic transport, a kinetic equation for the unbound molecules is obtained. This allows one to derive an expression for the low density limit of the transport diffusion coefficient D-0. In a second part of the paper the density dependence of D is studied following an Enskog-like approach, where the density dependence originates from the finite size of the molecules, combined with molecule-phonon relaxation. The density dependence is found to be the result of two effects: (i) the finite size introduces an additional contribution to the living force, resulting in an increase of D with increasing density, (ii) the collisions of the free with the bound molecules increase the momentum loss to the substrate and result in a decrease of D. The resulting expression for Das a function of the occupancy (theta) shows under reasonable assumptions for the kinetic parameters a dependence close to 1/(1 - theta), in agreement with the universal behavior observed in nanochannels of zeolites. (C) 1997 American Institute of Physics. [S0021-9606(97)51341-9].