  The recently-developed non-equilibrium extension of the self-consistent
generalized Langevin equation theory of irreversible relaxation [Phys. Rev. E
(2010) 82, 061503; ibid. 061504] is applied to the description of the
irreversible process of equilibration and aging of a glass-forming soft-sphere
liquid that follows a sudden temperature quench, within the constraint that the
local mean particle density remains uniform and constant. For these particular
conditions, this theory describes the non-equilibrium evolution of the static
structure factor S(k;t) and of the dynamic properties, such as the
self-intermediate scattering function F_S(k,\tau;t), where tau is the
correlation delay time and t is the evolution or waiting time after the quench.
Specific predictions are presented, for the deepest quench (to zero
temperature). The predicted evolution of the alpha-relaxation time
\tau_\alpha(t) as a function of t allows us to define the equilibration time
t^{eq}, as the time after which \tau_\alpha has attained its equilibrium value
\tau_\alpha^{eq}. It is predicted that both, t^{eq}(\phi) and
\tau_{\alpha}^{eq}, diverge as \phi \to \phi^{(a)}, where \phi^{(a)} is the
hard-sphere dynamic-arrest volume fraction \phi^{(a)}\ (\approx 0.582), thus
suggesting that the measurement of equilibrium properties at and above
\phi^{(a)} is experimentally impossible.
