Finite-size scaling and Monte Carlo simulations of first-order phase transitions

We develop a detailed finite-size-scaling theory at a general, asymmetric, temperature-driven, strongly first-order phase transition in a system with periodic boundary conditions. We compute scaling functions for various cumulants of energy in the form U(L,t)=U₀(tLᵈ)+L^-d{U}₁$(${tL}ᵈ$) with t=1-${T}c$/T. In particular, we consider the specific heat and Binder's fourth cumulant and show this has a minimum value of 2/3-(${e}₁$/${e}₂$-${e}₂$/${e}₁)²/12+O(L^-d) at a temperature Tc⁽²⁾(L)-Tc=O(L^-d). Various other pseudocritical temperatures corresponding to extrema of other cumulants are evaluated. We compare these theoretical predictions with extensive Monte Carlo simulations of the nominally strong first-order transitions in the eight- and ten-state Potts models in two dimensions for system sizes L{≤}50. The ten-state simulations agree with theory in all details in contrast to the eight-state data, and we give estimates for the bulk specific heats at Tc using all exactly known analytic results. A criterion is developed to estimate numerically whether or not system sizes used in a simulation of a first-order transition are in the finite-size-scaling regime.