Violent and mild relaxation of an isolated self-gravitating uniform and
  spherical cloud of particles
  The collapse of an isolated, uniform and spherical cloud of self-gravitating
particles represents a paradigmatic example of a relaxation process leading to
the formation of a quasi-stationary state in virial equilibrium. We consider
several N-body simulations of such a system, with the initial velocity
dispersion as a free parameter. We show that there is a clear difference
between structures formed when the initial virial ratio is b_0 =2K_0/W_0 <
b_0^c ~ -1/2 and b_0> b_0^c. These two sets of initial conditions give rise
respectively to a mild and violent relaxation occurring in about the same time
scale: however in the latter case the system contracts by a large factor, while
in the former it approximately maintains its original size. Correspondingly the
resulting quasi equilibrium state is characterized by a density profile
decaying at large enough distances as ~1/r^4 or with a sharp cut-off. The case
b_0<b_0^c can be well described by the Lynden-Bell theory of collisionless
relaxation considering the system confined in a box. On the other hand the
relevant feature for b_0>b_0^c is the ejection of particles and energy, which
is not captured by such a theoretical approach: for this case we introduce a
simple physical model to explain the formation of the power-law density
profile. This model shows that the behavior n(r) ~1/r^4 is the typical density
profile that is obtained when the initial conditions are cold enough that mass
and energy ejection occurs. In addition, we clarify the origin of the critical
value of the initial virial ratio b_0^c.
