Bringing discrete-time Langevin splitting methods into agreement with thermodynamics

TL;DR

This paper revises a Langevin splitting method to align with thermodynamic principles, introducing a new velocity concept that ensures correct statistical and transport properties in simulations, including quantum molecular dynamics.

Contribution

It establishes a link between ABO splitting and GJ methods, ensuring thermodynamic correctness and introduces a novel velocity for improved statistical accuracy.

Findings

GJ methods have at least weak second order accuracy.

The new half-step velocity yields correct kinetic and transport statistics.

Numerical simulations demonstrate the method's effectiveness in quantum molecular dynamics.

Abstract

In light of the recently published complete set of statistically correct Gronbech-Jensen (GJ) methods for discrete-time thermodynamics, we revise a differential operator splitting method for the Langevin equation in order to comply with the basic GJ thermodynamic sampling features, namely the Boltzmann distribution and Einstein diffusion, in linear systems. This revision, which is based on the introduction of time scaling along with flexibility of a discrete-time velocity attenuation parameter, provides a direct link between the ABO splitting formalism and the GJ methods. This link brings about the conclusion that any GJ method has at least weak second order accuracy in the applied time step. It further helps identify a novel half-step velocity, which simultaneously produces both correct kinetic statistics and correct transport measures for any of the statistically sound GJ methods. Explicit algorithmic expressions are given for the integration of the new half-step velocity into the GJ set of methods. Numerical simulations, including quantum-based molecular dynamics (QMD) using the QMD suite LATTE, highlight the discussed properties of the algorithms as well as exhibit the direct application of robust, time step independent stochastic integrators to quantum-based molecular dynamics.