  In this study, inspired by the work of K. Nakazato and K. Kitahara [Prog.
Theor. Phys. 64, 2261 (1980)], we consider the theoretical problem of tracer
particle diffusion in an environment of diffusing hardcore interacting crowder
particles. The tracer particle has a different diffusion constant from the
crowder particles. Based on a transformation of the generating function, we
provide an exact formal expansion for the tracer particle probability density,
valid for any lattice in the thermodynamic limit. By applying this formal
solution to dynamics on regular Bravais lattices we provide a closed form
approximation for the tracer particle diffusion constant which extends the
Nakazato and Kitahara results to include also b.c.c. and f.c.c. lattices.
Finally, we compare our analytical results to simulations in two and three
dimensions.
