  Molecular dynamics refers to the computer simulation of a material at the
atomic level. An open problem in numerical analysis is to explain the apparent
reliability of molecular dynamics simulations. The difficulty is that
individual trajectories computed in molecular dynamics are accurate for only
short time intervals, whereas apparently reliable information can be extracted
from very long-time simulations. It has been conjectured that long molecular
dynamics trajectories have low-dimensional statistical features that accurately
approximate those of the original system. Another conjecture is that numerical
trajectories satisfy the shadowing property: that they are close over long time
intervals to exact trajectories but with different initial conditions. We prove
that these two views are actually equivalent to each other, after we suitably
modify the concept of shadowing. A key ingredient of our result is a general
theorem that allows us to take random elements of a metric space that are close
in distribution and embed them in the same probability space so that they are
close in a strong sense. This result is similar to the Strassen-Dudley Theorem
except that a mapping is provided between the two random elements. Our results
on shadowing are motivated by molecular dynamics but apply to the approximation
of any dynamical system when initial conditions are selected according to a
probability measure.

Numerical Analysis;Dynamical Systems