(Keynote) Accurate Configurational and Kinetic Measures in Discrete-Time Langevin Dynamics
Authors: Jensen Lucas Frese Gronbech , Gronbech-Jensen Niels 
Published: February 2020

Numerical simulations of atomic and molecular ensembles by Molecular Dynamics always involve discretization of time, and as the time step is increased the discrete-time behavior becomes increasingly different from that of the anticipated continuous-time dynamics. This creates a dilemma for any simulation of a dynamical system: use a small time-step, resulting in dynamics that resemble the desired continuous-time behavior at the expense of computational efficiency; or use a large time step that makes the simulation finish sooner at the expense of accuracy. These are crucial considerations as many applications in Molecular Dynamics challenge the available computational resources either due to large spatial scales, large temporal scales (e.g., interfacial diffusion), or overwhelming atomic interactions (e.g., electrostatics). Adding to this dilemma is a much-overlooked fundamental problem with discrete-time dynamics; namely that the so-called velocity variable does, in fact, not produce the conjugate momentum variable to the simulated position in discrete time. Thus, the commonly used Verlet algorithm in Molecular Simulations, is not only producing errors that depend on the square of the simulated time step, it is further inducing those errors differently in velocity and position, such that self-consistency between kinetic and configurational measures is broken. Effects arising from this problem include that kinetic energy cannot be used as a precise indicator of configurational sampling temperature, and this causes a number of potentially severe issues if the time step is challenged. For example, isothermal ensembles that are simulated by methods, in which the measured kinetic temperature is used to control the thermodynamics, will inevitably lead to erroneous configurational sampling. Another example is that measures of diffusion, which can generally be made both by the square displacement and by velocity autocorrelation, are accurately measured only by square displacement in discrete time. It is therefore essential to understand the features of different algorithms such that optimal properties can be chosen for a given set of problems and objectives.
Our aim is here to improve the simulation techniques for systems in thermal equilibrium by creating an algorithm, which will provide the correct thermodynamic response regardless of the applied time step; for as long as the time step is within the stability limit. We briefly review the GJF stochastic Størmer-Verlet algorithm for the evolution of Langevin equations in a manner that preserves proper configurational sampling (diffusion and Boltzmann distribution) in discrete time. The resulting method, which is as simple as conventional Verlet schemes, has been numerically tested on both low-dimensional nonlinear systems as well as more complex molecular ensembles with many degrees of freedom. In light of the fundamental artifacts introduced by discrete time, we provide a simple intuitive picture of the unique benefits of our algorithm. We then present a statistical solution to the “velocity problem” by introducing a new, revised discrete-time velocity variable, which produces correct kinetic statistics in equilibrium. The new method is demonstrated to give exact statistics simultaneously for configurational and kinetic measures independently of the applied time step, and we demonstrate how these attractive features are useful for nonlinear and complex systems, such as Molecular Dynamics.