Tracer diffusion in lattice gases

It has been proved that a tracer particle in a reversible lattice gas converges to Brownian motion. However, only in a few particular cases has a strictly positive self-diffusion coefficient D been established. Here we supply the missing piece and show that D>0 in general. The exceptions are one-dimensional lattice gases with nearest neighbor jumps only, for which D=0. The proof establishes a variational formula for D which could be used to obtain realistic bounds.