  The statistical mechanics of N cold dark matter (CDM) particles interacting
via a softened gravitational potential is reviewed in the microcanonical
ensemble and mean-field limit. A phase diagram for the system is computed as a
function of the total energy E and gravitational softening length epsilon. For
softened systems, two stable phases exist: a collapsed phase, whose radial
density profile rho(r) is a central Dirac cusp, and an extended phase, for
which rho(r) has a central core and rho(r) ~ r^{-2.2} at large r. It is shown
that many N-body simulations of CDM haloes in the literature inadvertently
sample the collapsed phase only, even though this phase is unstable when there
is zero softening. Consequently, there is no immediate reason to expect
agreement between simulated and observed profiles unless the gravitational
potential is appreciably softened in nature.
