Optimum Monte Carlo Simulations: Some Exact Results

TL;DR

This paper provides exact analytical results for acceptance ratios and mean squared displacements in Monte Carlo simulations of harmonic oscillators, revealing dimension-independent properties and analyzing relaxation dynamics.

Contribution

It derives exact formulas for key Monte Carlo metrics and explores the spectral properties of the process, offering new insights into simulation efficiency and dynamics.

Findings

Acceptance ratio is independent of dimensionality when using uniform radial displacements.

Exact mean squared displacement values are obtained for the harmonic oscillator.

Spectral analysis accurately describes the relaxation time of the process.

Abstract

We obtain exact results for the acceptance ratio and mean squared displacement in Monte Carlo simulations of the simple harmonic oscillator in $D$ dimensions. When the trial displacement is made uniformly in the radius, we demonstrate that the results are independent of the dimensionality of the space. We also study the dynamics of the process via a spectral analysis and we obtain an accurate description for the relaxation time.