  It is well known that chaotic dynamic systems (such as three-body system,
turbulent flow and so on) have the sensitive dependance on initial conditions
(SDIC). Unfortunately, numerical noises (such as truncation error and round-off
error) always exist in practice. Thus, due to the SDIC, long-term accurate
prediction of chaotic dynamic systems is practically impossible, and therefore
numerical simulations of chaos are only mixtures of "true" solution with
physical meanings and numerical noises without physical meanings. However, it
is traditionally believed that statistic computations based on such kind of
"mixtures" of numerical simulations of chaotic dynamic systems are acceptable.
In this paper, using the so-called "clean numerical simulation" (CNS) whose
numerical noises might be much smaller even than micro-level physical
uncertainty and thus are negligible, we gain accurate prediction of a chaotic
dynamic system in a long enough interval of time. Then, based on these reliable
simulations, the influence of numerical noises on statistic computations is
investigated. It is found that the influence of numerical noises is negligible
when statistic results are time-independent. Unfortunately, when a chaotic
dynamic system is far from equilibrium state so that its statistics are
time-dependent, numerical noises have a great influence even on statistic
computations. It suggests that even the direct numerical simulations (DNS)
might not give reliable statistic computations for non-equilibrium dynamic
systems with the SDIC.
