  One of the fundamental aspects of statistical behaviour in many-body systems
is exponential divergence of neighbouring orbits, which is often discussed in
terms of Liapounov exponents. Here we study this topic for the classical
gravitational N-body problem. The application we have in mind is to old stellar
systems such as globular star clusters, where N~10^6, and so we concentrate on
spherical, centrally concentrated systems with total energy E<0. Hitherto no
connection has been made between the time scale for divergence (denoted here by
t_e) and the two-body relaxation time scale (t_r), even though both may be
calculated by consideration of two-body encounters.
  In this paper we give a simplified model showing that divergence in phase
space is initially roughly exponential, on a timescale proportional to the
crossing time (defined as a mean time for a star to cross from one side of the
system to another). In this phase t_e is much less than t_r, if N is not too
small (i.e. N is much more than about 30). After several e-folding times, the
model shows that the divergence slows down. Thereafter the divergence (measured
by the energies of the bodies) varies with time as t^{1/2}, on a timescale
nearly proportional to the familiar two-body relaxation timescale, i.e. t_e ~
t_r in this phase. These conclusions are illustrated by numerical results.
