NVU dynamics. I. Geodesic motion on the constant-potential-energy hypersurface

TL;DR

This paper introduces a stable, time-reversible algorithm for simulating geodesic motion on constant potential-energy hypersurfaces, validated on Lennard-Jones liquids, and compares it to traditional NVE methods.

Contribution

A novel, numerically stable NVU algorithm for geodesic simulation on constant potential-energy surfaces, with proven stability and equivalence to NVE in structural properties.

Findings

The NVU algorithm is numerically stable with smoothed force cutoff.

Potential-energy and step-length conservation are achieved in the modified algorithm.

NVU and NVE algorithms produce identical radial distribution functions.

Abstract

An algorithm is derived for computer simulation of geodesics on the constant potential-energy hypersurface of a system of N classical particles. First, a basic time-reversible geodesic algorithm is derived by discretizing the geodesic stationarity condition and implementing the constant potential energy constraint via standard Lagrangian multipliers. The basic NVU algorithm is tested by single-precision computer simulations of the Lennard-Jones liquid. Excellent numerical stability is obtained if the force cutoff is smoothed and the two initial configurations have identical potential energy within machine precision. Nevertheless, just as for NVE algorithms, stabilizers are needed for very long runs in order to compensate for the accumulation of numerical errors that eventually lead to "entropic drift" of the potential energy towards higher values. A modification of the basic NVU algorithm is introduced that ensures potential-energy and step-length conservation; center-of-mass drift is also eliminated. Analytical arguments confirmed by simulations demonstrate that the modified NVU algorithm is absolutely stable. Finally, simulations show that the NVU algorithm and the standard leap-frog NVE algorithm have identical radial distribution functions for the Lennard-Jones liquid.