  Self-gravitating Newtonian systems consisting of a very large number of
particles have generally defied attempts to describe them using statistical
mechanics. This is paradoxical since many astronomical systems, or simulations
thereof, appear to have universal, equilibrium structures for which no physical
basis exist. A decade ago we showed that extremizing the number of microstates
with a given energy per unit mass, under the constraints of conserved total
energy and mass, leads to the maximum entropy state, $n(E) \propto \exp
(-\beta(E-\Phi_0))-1$, known as DARKexp. This differential energy distribution,
and the resulting density structures, closely approximate those of dark-matter
halos with central cusps, $\rho \sim r^{-1}$, and outer parts, $\rho \sim
r^{-4}$. Here we define a non-equilibrium functional, $S_D$, which is maximized
for DARKexp and increases monotonically during the evolution towards
equilibrium of idealized collisionless systems of the Extended Spherical Infall
Model. Systems that undergo more mixing more closely approach DARKexp.
