  We present an equilibrium statistical mechanical theory of collisionless
self-gravitational systems with isotropic velocity distributions. Compared to
existing standard theories, we introduce two changes: (1) the number of
possible microstates is computed in energy (orbit) space rather than phase
space and (2) low occupation numbers are treated more appropriately than using
Stirling's approximation. Combined, the two modifications predict that the
relaxed parts of collisionless self-gravitating systems, such as dark-matter
halos, have a differential energy distribution N(E) ~ [exp(phi_0 - E) - 1],
dubbed "DARKexp". Such systems have central power-law density cusps rho(r) ~
r^-1, which suggests a statistical mechanical origin of cusps in simulated
dark-matter halos.
