# Accurate Configurational and Kinetic Statistics in Discrete-Time Langevin Systems

## TL;DR

This paper introduces a new velocity definition for the GJF thermostat in Langevin systems, enabling accurate, time-step-independent kinetic and configurational statistics in discrete-time simulations, including molecular dynamics.

## Contribution

A novel velocity variable for the GJF thermostat that provides correct kinetic responses and enables a new Leap-Frog algorithm for accurate statistical sampling.

## Key findings

- The new velocity yields correct kinetic energy and fluctuations.
- The algorithm is stable and accurate for nonlinear systems.
- It maintains time-step independence in complex molecular simulations.

## Abstract

We expand on the previously published Gr{\o}nbech-Jensen Farago (GJF) thermostat, which is a thermodynamically sound variation on the St{\o}rmer-Verlet algorithm for simulating discrete-time Langevin equations. The GJF method has been demonstrated to give robust and accurate configurational sampling of the phase space, and its applications to, e.g., Molecular Dynamics is well established. A new definition of the discrete-time velocity variable is proposed based on analytical calculations of the kinetic response of a harmonic oscillator subjected to friction and noise. The new companion velocity to the GJF method is demonstrated to yield correct and time-step-independent kinetic responses for, e.g., kinetic energy, its fluctuations, and Green-Kubo diffusion based on velocity autocorrelations. This observation allows for a new and convenient Leap-Frog algorithm, which efficiently and precisely represents statistical measures of both kinetic and configurational properties at any time step within the stability limit for the harmonic oscillator. We outline the simplicity of the algorithm and demonstrate its attractive time-step-independent features for nonlinear and complex systems through applications to a one-dimensional nonlinear oscillator and three-dimensional Molecular Dynamics.