Statistical ensembles of virialized halo matter density profiles
  We define and study statistical ensembles of matter density profiles
describing spherically symmetric, virialized dark matter haloes of finite
extent with a given mass and total gravitational potential energy. Our
ensembles include spatial degrees of freedom only, a microstate being a
spherically symmetric matter density function. We provide an exact solution for
the grand canonical partition functional, and show its equivalence to that of
the microcanonical ensemble. We obtain analytically the mean profiles that
correspond to an overwhelming majority of microstates. All such profiles have
an infinitely deep potential well, with the singular isothermal sphere arising
in the infinite temperature limit. Systems with virial radius larger than
gravitational radius exhibit a localization of a finite fraction of the energy
in the very center. The universal logarithmic inner slope of unity of the NFW
haloes is predicted at any mass and energy if an upper bound is set to the
maximal depth of the potential well. In this case, the statistically favored
mean profiles compare well to the NFW profiles. For very massive haloes the
agreement becomes exact.
