An exact solution of the spin-spin autocorrelation function for a one-dimensional system of hard rods

It is shown exactly that, in a one-dimensional system of hard rods with spins, the autocorrelation function of any function of spin F(w) decays as t^-1 at long times provided that <F>_eq exists and that g (0) ≠ 0, where g (v) is the linear velocity distribution function. As a consequence of this, when F(w) = w, the spin diffusion coefficient defined by the Kubo relation D_s = ∫₀∫ <w(0)w(t)> X d t does not exist. The results are true for arbitrary initial equilibrium velocity and spin distributions, the only restriction being that they be symmetric.