Two-particle approximation to the diffusion coefficient of a tracer particle

In this paper we show that the diffusion behaviour of a tracer particle in a lattice gas of hard-core particles at a given concentration may be approximated, for the whole concentration range, by the diffusion behaviour of a tracer particle in a two-particle system with effective transition rates. In this approach, the other particle is an effective background particle with a dual particle-vacancy character. The diffusion constant obtained in the present approximation is shown to be equivalent to the one obtained by solving a set of coupled many-body rate equations with a second-order approximation, as originally given by Tahir-Kheli and Elliott. Our solution relies exclusively on random-walk methods and we give, in this framework, the exact solution for the diffusion coefficient of a tracer particle in a two-particle system with arbitrary transition rates.