Correlated random walk on lattices: Tracer diffusion in one dimension

To test the accuracy of the two-point occupancy correlation function for a labeled particle hopping on uniform lattices (amongst an arbitrary concentration of hopping background particles) recently given by Tahir-Kheli and Elliott, we examine their solution for a one-dimensional, linear-chain lattice where the problem is particularly resistant to simple decoupling approximations. We find the theory performs better than we originally anticipated in the sense that even in one dimension it retains some of its correct features. As a by-product of this investigation, we obtain some new exact results valid in specific limits. In particular, for large values of the ratio, $\eta$, of the hopping rates, $J$ and $J^0$, of the background and tracer particles, the time dependence of the mean-square deviation of the tracer displays the following interesting behavior: $〈x^2\mathrm{}\left(t\right)\mathrm{}〉\to 2v{a}^2\left(J^{0}t\right)$, if $\eta \gg J^{0}t\gg 1$, while for very long times, i.e., $J^{0}t\gg \eta \gg 1$, $〈x^2\mathrm{}\left(t\right)\mathrm{}〉\to \left(\frac{2v{a}^2}{c}\right){\left(\frac{\mathrm{Jt}}{\pi }\right)}^{\frac{1}{2}}$.