  Self-gravitating system are non-equilibrium a priory. A new approach is
proposed, which employs a non-equilibrium statistical operator into account
inhomogeneous distribution of particles and temperature. The method involves
the saddle - point procedure to find the dominant contribution to the partition
function and thus to obtain all thermodynamic parameters of the system.
Probable peculiar features in the behavior of self-gravitating system are
considered for various condition. The equation of state for self-gravitating
system has been obtained. A new length of statistical instability and parameter
of the spatial inhomogeneous distribution of particles and temperature are
obtained for real gravitating system.

Statistical Mechanics