THE NOTION OF ERROR IN LANGEVIN DYNAMICS .1. LINEAR-ANALYSIS

The notion of error in practical molecular and Langevin dynamics simul ations of large biomolecules is far from understood because of the rel atively large value of the timestep used, the short simulation length, and the low-order methods employed. We begin to examine this issue wi th respect to equilibrium and dynamic time-correlation functions by an alyzing the behavior of selected implicit and explicit finite-differen ce algorithms for the Langevin equation. We derive: local stability cr iteria for these integrators; analytical expressions for the averages df the potential, kinetic, and total energy; and various limiting case s (e.g., timestep and damping constant approaching zero), for a system of coupled harmonic oscillators. These results are then compared to t he corresponding exact solutions for the continuous problem, and their implications to molecular dynamics simulations are discussed. New con cepts of practical and theoretical importance are introduced: scheme-d ependent perturbative damping and perturbative frequency functions. In teresting differences in the asymptotic behavior among the algorithms become apparent through this analysis, and two symplectic algorithms,' 'LIM2'' (implicit) and ''BBK'' (explicit), appear most promising on th eoretical grounds. One result of theoretical interest is that for the Langevin/implicit-Euler algorithm (''LI'') there exist timesteps for w hich there is neither numerical damping nor shift in frequency for a h armonic oscillator. However, this idea is not practical for more compl ex systems because these special timesteps can account only for one fr equency of the system,and a large damping constant is required. We the refore devise a more practical, delay-function approach to remove the artificial damping and frequency perturbation from LI. Indeed, a simpl e MD implementation for a system of coupled harmonic oscillators demon strates very satisfactory results in comparison with the velocity-Verl et scheme. We also define a probability measure to estimate individual trajectory error. This framework might be useful in practice for esti mating rare events, such as barrier crossing. To illustrate, this conc ept is applied to a transition-rate calculation, and transmission coef ficients for the five schemes are derived. (C) 1996 American Institute of Physics.