Surface diffusion coefficient near first-order phase transitions at low temperatures

We analyze the collective surface diffusion coefficient, $D_c$, near a first-order phase transition at which two phases coexist and the surface coverage, $\te$, drops from one single-phase value, $\te_+$, to the other one, $\te_-$. Contrary to other studies, we consider the temperatures that are sufficiently sub-critical. Using the local equilibrium approximation, we obtain, both numerically and analytically, the dependence of $D_c$ on the coverage and system size, $N$, near such a transition. In the two-phase regime, when $\te$ ranges between $\te_-$ and $\te_+$, the diffusion coefficient behaves as a sum of two hyperbolas, $D_c \approx A/N|\te - \te_-| + B/N|\te - \te_+|$. The steep hyperbolic increase in $D_c$ near $\te_\pm$ rapidly slows down when the system gets from the two-phase regime to either of the single-phase regimes (when $\te$ gets below $\te_-$ or above $\te_+$), where it approaches a finite value. The crossover behavior of $D_c$ between the two-phase and single-phase regimes is described by a rather complex formula involving the Lambert function. We consider a lattice-gas model on a triangular lattice to illustrate these general results, applying them to four specific examples of transitions exhibited by the model.