The Widom–Rowlinson mixture on a sphere: elimination of exponential slowing down atfirst-order phase transitions

Abstract

Computer simulations of first-order phase transitions using ‘standard’ toroidal boundaryconditions are generally hampered by exponential slowing down. This is partly due tointerface formation, and partly due to shape transitions. The latter occur when dropletsbecome large such that they self-interact through the periodic boundaries. On a sphericalsimulation topology, however, shape transitions are absent. We expect that by using anappropriate bias function, exponential slowing down can be largely eliminated. In thiswork, these ideas are applied to the two-dimensional Widom–Rowlinson mixture confinedto the surface of a sphere. Indeed, on the sphere, we find that the number ofMonte Carlo steps needed to sample a first-order phase transition does not increaseexponentially with system size, but rather as a power law , with α≈2.5, and V the system area. This is remarkably close to a random walk for whichαRW = 2. The benefit of this improved scaling behavior for biased sampling methods, such as theWang–Landau algorithm, is investigated in detail.