we attempted to calculate the self diffusion properties of vortices. However, it turned out to be numerically ambiguous and tricky to trace the trajectories of individual vortices. This is beacause individual vortex around a plaquette is defined in terms of phase angles and one probes the movement of individual vortices not directly but only indirectly through changes of phases, which, at times especially when multi-vortex motion occurs, makes it ambiguous to determine the original position of a vortex corresponding to a new neighboring vortex.

One way to overcome this difficulty was to map the UFXY model onto a LCG via Villain transformation [@villain], where the positive charges in the LCG correspond to the positive current vortices in UFXY models. One can readily probe the diffusive dynamics of charges without ambiguity in the LCG unlike the case of UFXY model. Hence we can investigate both the structural relaxation dynamics and self diffusion dynamics of individual vortices in LCG with relative ease.

With this advantage, we have numerically investigated the equilibrium relaxation dynamics and diffusion characteristics of LCG with charge density factor $f$ near $1-g \simeq 0.382$. We observe that for some range of temperatures above the first order transition, the equilibrium 