Miller [26] was the
first to show that small changes in the initial conditions of an *N*-body system
result in exponentially diverging solutions.
Lecar [22] co-ordinated a study between many researchers, each of whom
independently computed the solution to
an *N*-body problem with identical initial conditions.
They found that different algorithms and computers gave results in
which some measures differed by as much as 100%.
More recent work on the growth of errors includes Kandrup and Smith
[19], who showed that under a large range of parameters,
the time scale over
which small perturbations grow by a factor of *e* (Euler's constant,
), called the *e*-folding time, is
comparable to the *crossing time*
(the average time it takes a particle to cross the system once).
Goodman, Heggie, and Hut [10]
developed a detailed theory of the growth of
small perturbations, and verified it with
simulation to show that the exponential growth of small errors results
mostly from close encounters, which occur infrequently. This is an
interesting result because it says that, even though the full phase-space
solutions may experience exponential error growth,
this growth is due mainly to a
few particles that undergo close encounters. Perhaps the results of
collisionless systems can be trusted for longer than collisional ones, since
close encounters have a much smaller effect in the former.
Kandrup and Smith [21]
showed that as softening is increased (*i.e., * the collisionality
is decreased), the *e*-folding time grows slightly faster than linearly.
(*i.e., * the Lyapunov exponent decreases, so errors are magnified more slowly.)
They agree with Goodman *et al. *
that the error magnification is due more to the rare
individual particles whose errors grow much more quickly than
the average, although they claim the global potential also plays a role.
Presumably the particles whose errors grow more quickly than average
are ones that have suffered close encounters.

Since the time-scale for the growth of errors is so short (the
errors can be magnified by a factor of each crossing time),
the results of all *N*-body simulations may be suspect. If the
relative error per crossing time for a simulation is , then
after about
*p* crossing times, a particle's position will have an error comparable to
the size of the system -- in other words, all information will have been lost
about the particle's position.
Typical simulations today have a *p* between 4 and 8.

I am not
aware of *any* convincing justification for the belief that statistical
measures taken from these simulations are valid, although I tend to share the
same ``warm fuzzy'' feelings that most astronomers have -- namely that,
in some sense, large *N*-body simulations give valid statistical
results. The problem is, in *what* sense are they valid, if any;
*how* valid are they;
and finally, *how badly are we allowed to integrate before
validity is lost*?

Sun Dec 29 23:43:59 EST 1996

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