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The notion of error in Langevin dynamics. I. Linear analysis

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The notion of error in practical molecular and Langevin dynamics simulations
of large biomolecules is far from understood because of the relatively
large value of the timestep used, the short simulation length, and the
low-order methods employed. We begin to examine this issue with respect
to equilibrium and dynamic time-correlation functions by analyzing the
behavior of selected implicit and explicit finite-difference algorithms
for the Langevin equation. We derive: local stability criteria for these
integrators; analytical expressions for the averages of the potential,
kinetic, and total energy; and various limiting cases (e.g., timestep and
damping constant approaching zero), for a system of coupled harmonic oscillators.
These results are then compared to the corresponding exact solutions
for the continuous problem, and their implications to molecular dynamics
simulations are discussed. New concepts of practical and theoretical importance
are introduced: scheme-dependent perturbative damping and perturbative
frequency functions. Interesting 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 theoretical grounds. One result of theoretical interest is that
for the Langevin/implicit-Euler algorithm ("LI") there exist timesteps
for which there is neither numerical damping nor shift in frequency for
a harmonic oscillator. However, this idea is not practical for more complex
systems because these special timesteps can account only for one frequency
of the system, and a large damping constant is required. We therefore devise
a more practical, delay-function approach to remove the artificial damping
and frequency perturbation from LI. Indeed a simple MD implementation for
a system of coupled harmonic oscillators demonstrates very satisfactory
results in comparison with the velocity-Verlet scheme. We also define a
probability measure to estimate individual trajectory error. This framework
might be useful in practice for estimating rare events, such as barrier
crossing. To illustrate, this concept is applied to a transition-rate calculation,
and transmission coefficients for the five schemes are derived

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