Defining velocities for accurate kinetic statistics in the GJF thermostat

TL;DR

This paper investigates how to define velocities in the GJF thermostat to accurately measure kinetic energy, revealing limitations of on-site velocities and proposing multiple valid two-point velocity definitions for improved kinetic sampling.

Contribution

It provides a comprehensive analysis of velocity definitions in the GJF thermostat, identifying feasible two-point velocities and offering practical algorithms for accurate kinetic statistics.

Findings

On-site velocity cannot yield correct kinetic temperature independently of time step.

Multiple two-point velocity definitions can accurately measure kinetic energy.

Explicit formulas and algorithms for these velocities are provided.

Abstract

We expand on two previous developments in the modeling of discrete-time Langevin systems. One is the well-documented Gr{\o}nbech-Jensen Farago (GJF) thermostat, which has been demonstrated to give robust and accurate configurational sampling of the phase space. Another is the recent discovery that also kinetics can be accurately sampled for the GJF method. Through a complete investigation of all possible finite difference approximations to the velocity, we arrive at two main conclusions:~1) It is not possible to define a so-called on-site velocity such that kinetic temperature will be correct and independent of the time step, and~2) there exists a set of infinitely many possibilities for defining a two-point (leap-frog) velocity that measures kinetic energy correctly for linear systems in addition to the correct configurational statistics obtained from the GJF algorithm. We give explicit expressions for the possible definitions, and we incorporate these into convenient and practical algorithmic forms of the normal Verlet-type algorithms along with a set of suggested criteria for selecting a useful definition of velocity.