Diffusion in concentrated lattice gases. III. Tracer diffusion on a one-dimensional lattice

The dynamical process of the diffusion of tagged particles in a one-dimensional concentrated lattice gas is investigated. The particles are noninteracting except that double occupancy is forbidden. The mean-square displacement of a tagged particle is calculated for all times by an approximate theory and compared to results from Monte Carlo simulations. The overall agreement is quite good. For an infinite chain and for large time $t$ the mean-square displacement is found to increase proportionally to $t^{\frac{1}{2}}$ in agreement with existing results. For periodic chains it increases as $2D_{\mathrm{tr}}t$ for large times, with a coefficient of tracer diffusion $D_{\mathrm{tr}}$ inversely proportional to the number of particles on the chain. This, too, is in agreement with the results of older calculations. In the case of hard reflecting walls finally the mean-square displacement asymptotically approaches a constant, which can be calculated simply.