Is the broad histogram random walk dynamics correct?

TL;DR

This paper demonstrates that the traditional broad histogram single-spin-flip random walk dynamics fails to produce correct microcanonical averages due to violation of detailed balance, and proposes a modified dynamics that corrects this issue.

Contribution

The authors identify the flaw in the existing broad histogram dynamics and introduce a modified version that ensures detailed balance and accurate microcanonical averages.

Findings

Original dynamics violates detailed balance, especially in small systems.

Modified dynamics restores detailed balance and produces a flat histogram.

The new method provides correct microcanonical averages across system sizes.

Abstract

We show explicitly that the broad histogram single-spin-flip random walk dynamics does not give correct microcanonical average even in one dimension. The dynamics violates detailed balance condition by an amount which decreases with system size. As a result, in distribution different configurations with the same energy can have different probabilities. We propose a modified dynamics which ensures detailed balance and the histogram obtained from this dynamics is exactly flat.