Isotropic-Nematic Phase Transitions in Gravitational Systems II: Higher
  Order Multipoles
  The gravitational interaction among bodies orbiting in a spherical potential
leads to the rapid relaxation of the orbital planes' distribution, a process
called vector resonant relaxation. We examine the statistical equilibrium of
this process for a system of bodies with similar semimajor axes and
eccentricities. We extend the previous model of Roupas et al. (2017) by
accounting for the multipole moments beyond the quadrupole, which dominate the
interaction for radially overlapping orbits. Nevertheless, we find no
qualitative differences between the behavior of the system with respect to the
model restricted to the quadrupole interaction. The equilibrium distribution
resembles a counterrotating disk at low temperature and a spherical structure
at high temperature. The system exhibits a first order phase transition between
the disk and the spherical phase in the canonical ensemble if the total angular
momentum is below a critical value. We find that the phase transition erases
the high order multipoles, i.e. small-scale structure in angular momentum
space, most efficiently. The system admits a maximum entropy and a maximum
energy, which lead to the existence of negative temperature equilibria.
