Time-dependent correlations between a pair of distinguishable particles hopping on a uniform one-dimensional chain

We derive equations for the time-dependent correlation function describing the relative motion of a pair of distinguishable particles hopping on a one-dimensional linear chain with all sites equivalent. This correlation function is directly proportional to the magnetization relaxation rate (T₁^-1) in NMR experiments when the relaxation is due to the dipolar interaction between hopping spins. The equations are exact in the limit where the concentration of vacancies on the chain is small and, we believe, yield qualitatively good results for other concentrations. In the limit where the frequency ω is much less than the hopping rate we predict that the correlation function and thus T₁^-1 are proportional to ω^-3/4. For three-dimensional systems the relaxation rate is frequency independent in this limit while crude (and incorrect) one-dimensional arguments lead to a ω^-1/2 dependence.