  Simulated dark matter profiles are often modelled as a `NFW' density profile
rather than a single power law. Recently, attention has turned to the rather
rigorous power-law behaviour exhibited by the `pseudo phase-space density' of
the dark matter halo, which is defined dimensionally in terms of the local
density and velocity dispersion of the dark matter particles. The non-power-law
behaviour of the density profile is generally taken to exclude simple
scale-free, in-fall models; however the power-law behaviour of the
`pseudo-density' is a counter indication. We argue in this paper that both
behaviours may be at least qualitatively understood in terms of a dynamically
evolving self-similarity, rather than the form for self-similar infall that is
fixed by cosmological initial conditions. The evolution is likely due to
collective relaxation such as that provided by the radial-orbit instability on
large scales. We deduce, from a distribution function given by first order
coarse-graining, both the NFW-type density profile and the power-law
pseudo-density profile. The results are not greatly sensitive to variation
about 3 in the power of the velocity dispersion used in the definition of the
phase space pseudo-density. We suggest that the power 2 may create the more
physical quantity, whose deviations from a power-law are a diagnostic of
incomplete relaxation.
