  In this paper, a quantitative characterization for the evolutionary sequence
of stellar self-gravitating system is investigated, focusing on the
pre-collapse stage of the long-term dynamical evolution. In particular, we
consider the quasi-equilibrium behaviors of the N-body systems in the setup of
the so-called Antonov problem, i.e., self-gravitating N-body system confined in
an adiabatic wall and try to seek a possible connection with thermostatistics
of self-gravitating systems. For this purpose, a series of long-term N-body
simulations with various initial conditions are performed. We found that a
quasi-equilibrium sequence away from the thermal equilibrium can be
characterized by the one-parameter family of the stellar models. Especially,
the stellar polytropic distribution satisfying the effective equation of state
$P\propto\rho^{1+1/n}$ provides an excellent approximation to the evolutionary
sequence of the N-body system. Based on the numerical results, we discuss a
link between the quasi-equilibrium state and the generalized thermostatistics
by means of the non-extensive entropy.
