Fast, hierarchical, and adaptive algorithm for Metropolis Monte Carlo simulations of long-range interacting systems

TL;DR

This paper introduces a fast, hierarchical, and adaptive Metropolis Monte Carlo algorithm for simulating long-range interacting systems, achieving significant speedups while maintaining exact dynamics, enabling studies of larger systems.

Contribution

The authors develop a novel algorithm that reduces computational complexity to O(N log N) for long-range systems, with exact dynamics preservation and practical speedup over traditional methods.

Findings

Achieves average complexity O(N log N) with small prefactors

Speedup factors larger than 10^4 in practical simulations

Applicable to large systems previously computationally infeasible

Abstract

We present a fast, hierarchical, and adaptive algorithm for Metropolis Monte Carlo simulations of systems with long-range interactions that reproduces the dynamics of a standard implementation exactly, i.e., the generated configurations and consequently all measured observables are identical, allowing in particular for nonequilibrium studies. The method is demonstrated for the power-law interacting long-range Ising model with nonconserved order parameter and a Lennard-Jones system both in two dimensions. The measured runtimes support an average complexity $O(N\log N)$, where $N$ is the number of spins or particles. Importantly, prefactors of this scaling behavior are small, which in practice manifests in speedup factors larger than $10^4$. The method is general and will allow the treatment of large systems that were out of reach before, likely enabling a more detailed understanding of physical phenomena rooted in long-range interactions.