The Widom-Rowlinson mixture on a sphere: Elimination of exponential slowing down at first-order phase transitions

Computer simulations of first-order phase transitions using standard toroidal boundary conditions are generally hampered by exponential slowing down. This is partly due to interface formation, and partly due to shape transitions. The latter occur when droplets become large such that they self-interact through the periodic boundaries. On a spherical simulation topology, however, shape transitions are absent. By using an appropriate bias function, we expect that exponential slowing down can be largely eliminated. In this work, these ideas are applied to the two-dimensional Widom-Rowlinson mixture confined to the surface of a sphere. Indeed, on the sphere, we find that the number of Monte Carlo steps needed to sample a first-order phase transition does not increase exponentially with system size, but rather as a power law $\tau \propto V^\alpha$, with $\alpha \approx 2.5$, and $V$ the system area. This is remarkably close to a random walk for which $\alpha$ equals 2. The benefit of this improved scaling behavior for biased sampling methods, such as the Wang-Landau algorithm, is investigated in detail.