On the Definition of Velocity in Discrete-Time, Stochastic Langevin Simulations

TL;DR

This paper develops new velocity measures for Langevin simulations that improve statistical accuracy and consistency with configurational coordinates, enhancing molecular dynamics sampling methods.

Contribution

It introduces and analyzes new velocity definitions compatible with Verlet-type algorithms, ensuring correct statistical features and practical implementation in simulations.

Findings

New velocities exhibit correct drift and Maxwell-Boltzmann distribution.

Half-step velocities are generally necessary for accurate statistical sampling.

Validated algorithms demonstrate improved statistical accuracy in simulations.

Abstract

We systematically develop beneficial and practical velocity measures for accurate and efficient statistical simulations of the Langevin equation with direct applications to computational statistical mechanics and molecular dynamics sampling. Recognizing that the existing velocity measures for the most statistically accurate discrete-time Verlet-type algorithms are inconsistent with the simulated configurational coordinate, we seek to create and analyze new velocity companions that both improve existing methods as well as offer practical options for implementation in existing computer codes. The work is based on the set of GJ methods that, of all methods, for any time step within the stability criteria correctly reproduces the most basic statistical features of a Langevin system; namely correct Boltzmann distribution for harmonic potentials and correct transport in the form of drift and diffusion for linear potentials. Several new and improved velocities exhibiting correct drift are identified, and we expand on an earlier conclusion that, generally, only half-step velocities can exhibit correct, time-step independent Maxwell-Boltzmann distributions. Specific practical and efficient algorithms are given in familiar forms, and these are used to numerically validate the analytically derived expectations. One especially simple algorithm is highlighted, and the ability of one of the new on-site velocities to produce statistically correct averages for a particular damping value is specified.