Statistical mechanics of self-gravitating systems: mixing as a criterion
  for indistinguishability
  We propose an association between the phase-space mixing level of a
self-gravitating system and the indistinguishability of its constituents (stars
or dark matter particles). This represents a refinement in the study of systems
exhibiting incomplete violent relaxation. Within a combinatorial analysis
similar to that of Lynden-Bell, we make use of this association to obtain a
distribution function that deviates from the Maxwell-Boltzmann distribution,
increasing its slope for high energies. Considering the smallness of the
occupation numbers for large distances from the center of the system, we apply
a correction to Stirling's approximation which increases the distribution slope
also for low energies. The distribution function thus obtained presents some
resemblance to the "S" shape of distributions associated with cuspy density
profiles (as compared to the distribution function obtained from the Einasto
profile), although it is not quite able to produce sharp cusps. We also argue
how the association between mixing level and indistinguishability can provide a
physical meaning to the assumption of particle-permutation symmetry in the
N-particle distribution function, when it is used to derive the one-particle
Vlasov equation, which raises doubts about the validity of this equation during
violent relaxation.
