  We study analytically the coarse and fine-grained distribution function
established by the self-similar infall of collisionless matter. We find this
function explicitly for isotropic and spherically symmetric systems in terms of
cosmological initial conditions. The coarse-grained function is structureless
and steady but the familiar phase space sheet sub-structure is recovered in the
fine-grained limit. By breaking the self-similarity of the halo infall we are
able to argue for a central density flattening. In addition there will be an
edge steepening. The best fitting analytic density function is likely to be
provided by a high order polytrope fit smoothly to an outer power law of index
-3 for isolated systems. There may be a transition to a -4 power law in the
outer regions of tidally truncated systems.
  We find a progressive central flattening that is expected to end either in
the non-singular isothermal sphere, or in non-singular metastable polytropic
cores. Therefore a collisionless system may pass through a family of polytropes
of increasing order, finally approaching the limit of the non-singular
isothermal sphere, if the `violent' collective relaxation is frequently
re-excited by `merger' events. Our results suggest that no physics beyond that
of the moderate collective relaxation often known as `violent relaxation' (due
we think to wave-particle scattering) is necessary to explain the nature of
dark matter density profiles.
