Bringing discrete-time Langevin splitting methods into agreement with thermodynamics

Area of Science:

• Computational Physics
• Statistical Mechanics
• Numerical Analysis

Background:

• The Grønbech-Jensen (GJ) methods provide statistically correct sampling for discrete-time thermodynamics.
• Existing differential operator splitting methods for the Langevin equation require revision to incorporate GJ thermodynamic sampling features.
• Linear systems in discrete-time thermodynamics necessitate methods that ensure Boltzmann distribution and Einstein diffusion.

Purpose of the Study:

• To revise a differential operator splitting method for the Langevin equation to comply with GJ thermodynamic sampling features.
• To establish a direct link between the ABO splitting formalism and GJ methods.
• To introduce a novel half-step velocity for improved kinetic statistics and transport measures.

Main Methods:

• Revision of a differential operator splitting method by introducing time scaling and a discrete-time velocity attenuation parameter.
• Development of explicit algorithmic expressions for integrating a novel half-step velocity into GJ methods.
• Numerical simulations, including quantum-based molecular dynamics (QMD) using the QMD suite Los Alamos Transferable Tight-Binding for Energetics (T4e).

Main Results:

• The revised method ensures compliance with Boltzmann distribution and Einstein diffusion in linear systems.
• A direct link between the ABO splitting formalism and GJ methods is established.
• Any GJ method is shown to possess at least weak second-order accuracy in the applied time step.
• A novel half-step velocity is identified, yielding correct kinetic statistics and transport measures.
• Robust, time-step-independent stochastic integrators are demonstrated for QMD.

Conclusions:

• The revised differential operator splitting method successfully integrates GJ thermodynamic sampling features.
• The identified half-step velocity offers a significant improvement for discrete-time thermodynamic simulations.
• The developed algorithms are applicable to quantum-based molecular dynamics, offering robust and time-step-independent integration.