Diffusion in infinite and semi-infinite lattices with long-range coupling

We prove that for a one-dimensional, infinite periodic lattice, with long-range coupling among sites, the diffusion of an initial delta-like pulse in the bulk is ballistic at all times, with a 'speed' that depends on the 'smoothness' of the dispersion relation. We obtain a closed-form expression for the mean square displacement (MSD), and show some relevant examples including finite-range coupling, exponentially decreasing coupling and power-law decreasing coupling. For the case of an initial excitation at the edge of the lattice, we find an approximate expression for the MSD that predicts ballistic behavior at long times in agreement with numerical results.