(Invited) On the Definition of Velocity in Statistically Accurate, Discrete-Time Simulations of Thermodynamics
Authors: Grønbech-Jensen Niels 
Published: August 2024

One of the most basic struggles in simulations of statistical and dynamical systems is how to appropriately balance the desire for simulation accuracy, obtained in the small time step limit, with simulation efficiency, obtained for large time steps. Obviously, we would like to have the benefits of both limits of accuracy and efficiency. Thus, understanding the influence of discrete time on the behavior of equations of motion is crucial for the understanding and optimization of numerical simulations in physical science and engineering. There are two different types of crucial discretization errors. One is that the simulated, discrete-time trajectory increasingly deviates from the true, continuous-time trajectory as the time step is increased. Another is that the simulated discrete-time velocity increasingly deviates from the expected velocity of the corresponding simulated trajectory as the time step is increased. While the former error is a signature that the simulated trajectory is not exactly the intended one, the simulation may still be representative of the true system. In contrast, the latter error is a signature of a fundamental internal inconsistency in the simulation, which may greatly complicate physical interpretation of the simulation results, especially when both configurational and kinetic measures are required. It is therefore crucially important to understand the relationship between a discrete-time coordinate and its corresponding velocity.
It has previously been shown that, unlike in continuous-time, discrete-time configurational dynamics does not need a discrete-time velocity variable. This realization allowed for the derivation of the complete configurational GJ set of stochastic Verlet-type algorithms for simulating the Langevin equation. This set has the key feature of simultaneously giving exact, time-step independent statistical sampling of damped, noisy harmonic oscillators, and giving exact spatial transport values, drift and Einstein diffusion, on flat surfaces. The apparent fact that a discrete-time velocity is not required to simulate a coordinate trajectory implies that discrete-time velocity is an ambiguous concept, which, in turn, implies that multiple velocity definitions can be entertained in order to avoid internal simulation inconsistencies between configurational and kinetic measures. From the derivation of the stochastic GJ methods it was shown that a statistically correct velocity obtained at the same time as the coordinate (an on-site velocity) cannot be obtained. In contrast, a velocity at the half-step between consecutive configurational time steps (a half-step velocity) was identified to produce exact statistical kinetic response for harmonic oscillators regardless of the applied time step. However, only for a single GJ method does this velocity also give the correct drift value.
We here derive new, practical velocity definitions that are optimized for statistical kinetic measures, such that, e.g., measures of kinetic energy and correlations with the configurational coordinate are exact for the noisy harmonic oscillator, while also producing the correct kinetic measure of transport on flat surfaces. We identify and analyze three half-step velocity definitions that fulfill these criteria, and we outline simple and practical Verlet-type algorithms that incorporate these velocities within the statistically desirable GJ format that provides the optimized configurational statistics. The application of the methods are demonstrated through characteristic Molecular Dynamics simulations of both simple and complex systems, illustrating how the resulting methods are able to efficiently and simultaneously simulate configurational and kinetic statistical ensembles in different situations using very large time steps.