Longer Time Steps for Molecular Dynamics

In molecular dynamics one solves Newton''s equations of motion. This is a challenging problem because the number of particles is large (thousands of atoms) but more importantly because stability severely limits the length of time steps relative to the total length needed for simulations---time steps are in the order of femtoseconds ($10^{-15}$ seconds) whereas simulations of a few microseconds ($10^{-6}$ seconds) up to one second are most desired. This situation also holds for multiple time stepping (MTS) integrators which evaluate different parts of the force at different frequencies. This work is an attempt to construct MTS integrators which are stable for long time steps. This has been done by applying a seminal idea that appeared in ~\cite{GaSS98}: modify the potential by defining it at time-averaged positions $\mathcal{A}(x)$, where the time averaging $\mathcal{A}(x)$ takes into account the high frequency vibrational motion. The resulting family of numerical integrators is called the mollified impulse method or MOLLY. Several averagings suggested, but not tested, in~\cite{GaSS98} are evaluated in this dissertation, along with more successful new ones. Finally, an efficient implementation of these averagings is devised. One of them is included in a production molecular dynamics code (NAMD 2). It is shown that the limitations in current MOLLY methods stem from a deficient representation of the eigenspace of the fast part of the forces. An alternative, less rigorous, approach based on ``Langevin stabilization'''' is explored. A mild damping and random noise is added to the system to remove instabilities without significantly affecting its dynamics.