Efficiency of Rejection-free dynamic Monte Carlo methods for homogeneous spin models, hard disk systems, and hard sphere systems

Abstract

We construct asymptotic arguments for the relative efficiency of rejection-free Monte Carlo (MC) methods compared to the standard MC method. We find that the efficiency is proportional to $\exp{({const} β)}$ in the Ising, $\sqrtβ$ in the classical XY, and $β$ in the classical Heisenberg spin systems with inverse temperature $β$, regardless of the dimension. The efficiency in hard particle systems is also obtained, and found to be proportional to $(\rhoc -ρ)^{-d}$ with the closest packing density $\rhoc$, density $ρ$, and dimension $d$ of the systems. We construct and implement a rejection-free Monte Carlo method for the hard-disk system. The RFMC has a greater computational efficiency at high densities, and the density dependence of the efficiency is as predicted by our arguments.