Self-diffusion in simple models: Systems with long-range jumps

We review some exact results for the motion of a tagged particle in simple models. Then, we study the density dependence of the self-diffusion coefficientD_N() in lattice systems with simple symmetric exclusion in which the particles can jump, with equal rates, to a set ofN neighboring sites. We obtain positive upper and lower bounds onF_N()=N{(1–)–[D_N()/D_N(0)]}/[(1–)]x for [0, 1]. Computer simulations for the square, triangular, and one-dimensional lattices suggest thatF_N becomes effectively independent ofN forN20.