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

TL;DR

This paper analyzes and demonstrates the efficiency advantages of rejection-free Monte Carlo methods over standard methods across various spin models and hard particle systems, especially at high densities and low temperatures.

Contribution

It provides asymptotic efficiency estimates for rejection-free Monte Carlo methods in different models and implements a new RFMC for hard-disk systems showing improved high-density performance.

Findings

Efficiency scales as exp(const * β) in Ising models

Efficiency scales as √β in XY models

Efficiency scales as β in Heisenberg models and as (ρc - ρ)^(-d) in hard particle 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 $\exp{({const} \beta)}$ in the Ising, $\sqrt{\beta}$ in the classical XY, and $\beta$ in the classical Heisenberg spin systems with inverse temperature $\beta$, regardless of the dimension. The efficiency in hard particle systems is also obtained, and found to be proportional to $(\rhoc -\rho)^{-d}$ with the closest packing density $\rhoc$, density $\rho$, 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.