  The thermodynamic behaviour of self-gravitating $N$-body systems has been
worked out by borrowing a standard method from Molecular Dynamics: the time
averages of suitable quantities are numerically computed along the dynamical
trajectories to yield thermodynamic observables. The link between dynamics and
thermodynamics is made in the microcanonical ensemble of statistical mechanics.
The dynamics of self-gravitating $N$-body systems has been computed using two
different kinds of regularization of the newtonian interaction: the usual
softening and a truncation of the Fourier expansion series of the two-body
potential. $N$ particles of equal masses are constrained in a finite three
dimensional volume. Through the computation of basic thermodynamic observables
and of the equation of state in the $P - V$ plane, new evidence is given of the
existence of a second order phase transition from a homogeneous phase to a
clustered phase. This corresponds to a crossover from a polytrope of index
$n=3$, i.e. $p=K V^{-4/3}$, to a perfect gas law $p=K V^{-1}$, as is shown by
the isoenergetic curves on the $P - V$ plane. The dynamical-microcanonical
averages are compared to their corresponding canonical ensemble averages,
obtained through standard Monte Carlo computations. A major disagreement is
found, because the canonical ensemble seems to have completely lost any
information about the phase transition. The microcanonical ensemble appears as
the only reliable statistical framework to tackle self-gravitating systems.
Finally, our results -- obtained in a ``microscopic'' framework -- are compared
with some existing theoretical predictions -- obtained in a ``macroscopic''
(thermodynamic) framework: qualitative and quantitative agreement is found,
with an interesting exception.
