Bias in Molecular Dynamics Averages due to Finite Step Size

Even if our model for the molecular system is exact, computational resources limit the simu-lation length, which introduces statistical error when computing ensemble averages. Additionally, molecular dynamics, unlike Monte Carlo, simulations give biased results due to the use of numer-ical integrators. This applies whether the dynamics is deterministic or stochastic; this discussion considers only the deterministic case. In molecular dynamics (MD) it is assumed that time averages of “physically meaningful func-tions ” do not depend on a detailed description of the initial conditions, which is a consequence of the ergodic hypothesis. Due to the chaotic nature of the underlying equations of motion, it is impossible to construct an exact trajectory over a long time interval. Hence, it is remarkable that there exist numerical integrators that can approximately preserve the correct invariant density for very long durations, in particular, symplectic integrators such as Verlet. There are three ways in which temporal discretization introduces bias into averages. First, the finite step size, ∆t, modifies the dynamics in a systematic way, with an effect that can be expressed as an asymptotic expansion in powers of ∆t. Second, discretization may affect ergodicity [10], which is particularly problematic since it cannot be easily detected or corrected. Third, it introduce