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Center for Nonlinear Studies |
A basic struggle in simulations of statistical and dynamical systems is how to appropriately balance simulation accuracy for small time steps with simulation efficiency for large ones. Thus, understanding the influence of discrete time on the behavior of equations of motion is crucial for the understanding and optimization of physical system simulations. We argue that, in computational statistical mechanics, 1) it is not necessary to obtain accurate trajectories in order to generate accurate statistics, and 2) a numerical method should first and foremost be analyzed by its configurational properties since momentum is an unnecessary quantity for discrete-time sampling of the phase-space [1,2]. Building on a derivation of the complete set of optimal stochastic Verlet-type integrators [3], we here provide a linear framework for analyzing the quality of the large number of stochastic integrators that have been proposed over the past five decades [4]. With some redundancy of logic we conclude that the previously identified complete set of integrators is the only set that allows for large time-step simulations, while preserving statistical accuracy in the most basic measures of diffusion, drift, and sampling (Boltzmann) distribution, even if the simulated trajectories suffer from time-step errors. The methods are remarkably simple and is implemented into existing codes, such as the Molecular Dynamics suite, LAMMPS.
[1] Grønbech-Jensen, Molecular Physics 118, e1662506 (2020)
[2] Grønbech-Jensen, Journal of Statistical Physics 191, 137 (2024)
[3] Finkelstein et al., Journal of Chemical Physics 153, 134101 (2020)
[4] Grønbech-Jensen, arXiv:2505.04100 (2025)
Host: Josh Finkelstein, T-1