  We perform Brownian dynamics simulations for studying the self-diffusion in
two-dimensional (2D) dusty plasma liquids, in terms of both mean-square
displacement and velocity autocorrelation function (VAF). Super-diffusion of
charged dust particles has been observed to be most significant at infinitely
small damping rate $\gamma$ for intermediate coupling strength, where the
long-time asymptotic behavior of VAF is found to be the product of $t^{-1}$ and
$\exp{(-\gamma t)}$. The former represents the prediction of early theories in
2D simple liquids and the latter the VAF of a free Brownian particle. This
leads to a smooth transition from super-diffusion to normal diffusion, and then
to sub-diffusion with an increase of the damping rate. These results well
explain the seemingly contradictory scattered in recent classical molecular
dynamics simulations and experiments of dusty plasmas.
