  We present an analysis of the behaviour of the `coarse-grained'
(`mesoscopic') rank partitioning of the mean energy of collections of particles
composing virialized dark matter halos in a Lambda-CDM cosmological simulation.
We find evidence that rank preservation depends on halo mass, in the sense that
more massive halos show more rank preservation than less massive ones. We find
that the most massive halos obey Arnold's theorem (on the ordering of the
characteristic frequencies of the system) more frequently than less massive
halos. This method may be useful to evaluate the coarse-graining level (minimum
number of particles per energy cell) necessary to reasonably measure signatures
of `mesoscopic' rank orderings in a gravitational system.
