  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). This effect
has been noted for some time in simulations, although individual particle
energies and their phase-space variables mix considerably. The present
investigation aimed to map ERP in terms of parameters involving the collective
range in which it is effective, as well as in terms of global and historical
characterisations of gravitational systems evolving towards equilibrium. We
examined a subset of the IllustrisTNG cosmological magnetohydrodynamical
simulations (TNG50-4 and TNG100-3), considering both their full and dark-only
versions. For each simulation, we selected the $20$ most massive haloes at
redshift $z=0$, tracing their ERP fractions back at selected redshift markers
({\tt z} $= \{1.0, 5.0, 10.0 \}$), and for a coarse-graining set ranging from
$5$ to $30$ energy bins. At the redshift marker {\tt z} $= 1$, we found high
ERP fractions (above $\sim 80 \%$) in both simulations, regardless of the
coarse-graining level. The {\it decline} in ERP fractions with redshift was
roughly a function of mass and fractional mass increase in the analysed TNG50-4
haloes, but not in the TNG100-3 ones, indicating a possible relative
susceptibility of the ERP effect to mass accretion for haloes less massive than
$\sim 10^{14} ~ M_{\odot}$. We confirmed earlier indications in the literature
concerning a possible "mesoscopic" constraint operative in a time span of at
least several Gyr.
