The challenge of stochastic Størmer-Verlet thermostats generating correct statistics

Area of Science:

• Computational physics
• Statistical mechanics
• Molecular dynamics

Background:

• Stochastic, discrete-time Størmer-Verlet algorithms are essential for simulating Langevin equations.
• Recent advancements have introduced a complete set of single random variable algorithms (GJ set).
• Outstanding questions remain regarding the benefits of multiple random variables and the selection of optimal algorithms.

Purpose of the Study:

• To investigate the algorithmic and statistical benefits of using multiple random variables per time step in Størmer-Verlet algorithms.
• To determine objective criteria for selecting among statistically correct algorithms for Langevin simulations.
• To analyze friction-induced differences in method stability and temporal scaling.

Main Methods:

• A general form for discrete-time equations with two random variables was analyzed using the GJ methodology.
• Thermodynamic correctness was enforced in linear systems to validate the algorithms.
• Numerical simulations of complex molecular systems and analytic considerations were employed.

Main Results:

• Correct configurational Boltzmann sampling and free-particle diffusion require identical random variables per time step.
• The GJ set encompasses all stochastic Størmer-Verlet methods yielding time step-independent statistics for linear systems.
• Friction-dependent discrete-time scaling differences were observed among methods, impacting stability.

Conclusions:

• Using identical random variables per time step is necessary for accurate statistical simulations of Langevin equations.
• The GJ-I/GJF-2GJ method is recommended due to its straightforward temporal scaling interpretation and statistical robustness.
• Algorithm selection should consider method-specific temporal scaling and stability in complex systems.