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

Henrik Christiansen

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 and XY spin models with nonconserved order parameter and a Lennard-Jones particle system, all in
two dimensions. The measured run times support an average complexity OðN log NÞ, 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.