Mesoscopic Energy Ranking Constraints in the IllustrisTNG Simulations
  We revisited the problem of "mixing" in a gravitational N-body system from
the point of view of the ordering of coarse-grained cells in the one-particle
energy space, here denoted {\it energy ranking preservation} (ERP) analysis. We
investigated a subset of the IllustrisTNG cosmological magnetohydrodynamical
simulations, considering both their full and dark-only versions. For each
simulation, we extracted data from the $4$ most massive ("reference") FOF
haloes at redshift $z=0$, which were analysed separately. The particle energies
in those haloes were sorted and partitioned into bins in terms of quintiles and
deciles. We then identified such particles at higher redshifts, considering
only if they were residing in FOF haloes, and assigned them to the same
initially defined energy bins. We found evidence of ERP, monotonically
declining as a function of redshift, reference halo mass and particle type. Our
results confirm other indications in the literature of a "mesoscopic"
constraint, operating at a collective level in the energy space, extending now
this effect also to baryons. In a coarse-grained sense, our results indicate
that, roughly, the most (less) gravitationally bounded masses today were
probably the most (less) bounded ones at redshifts even as high as $z \sim 5$.
Our results suggest a relation between the level of ERP and the time since a
major merger.
