Equilibrium Times for the Multicanonical Method

TL;DR

This paper introduces a new criterion to measure the equilibrium time of Markov process-based sampling methods, demonstrating its effectiveness on the 2D-Ising model and revealing power-law relationships and critical slowing down.

Contribution

It proposes a novel, general method to determine equilibrium time in Markov process sampling, validated on the 2D-Ising system with consistent results.

Findings

Power-law relationship between equilibrium time and system size

Power-law relationship between equilibrium time and energy levels

Observation of critical slowing down near the critical energy

Abstract

This work measures the time to equilibrium for the multicanonical method on the 2D-Ising system by using a new criterion, proposed here, to find the time to equilibrium, teq, of any sampling procedure based on a Markov process. Our new procedure gives the same results that the usual one, based on the magnetization, for the canonical Metropolis sampling on a 2D-Ising model at several temperatures. For the multicanonical method we found a power-law relationship with the system size, L, of teq=0.27(15) L^2.80(13), and with the number of energy levels to explore, kE, of teq=0.7(13) kE^1.40(11), in perfect agreement with the result just above. In addition, some kind of critical slowing down was observed around the critical energy. Our new procedure is completely general, and can be applied to any sampling method based on a Markov process.