Thermodynamics of gravitational clustering phenomena: $N$-body
  self-gravitating gas on the sphere $\mathbb{S}^{3}\subset\mathbb{R}^{4}$
  This work is devoted to the thermodynamics of gravitational clustering, a
collective phenomenon with a great relevance in the $N$-body cosmological
problem. We study a classical self-gravitating gas of identical
non-relativistic particles defined on the sphere $\mathbb{S}^{3}\subset
\mathbb{R}^{4}$ by considering gravitational interaction that corresponds to
this geometric space. The analysis is performed within microcanonical
description of an isolated Hamiltonian system by combining continuum
approximation and steepest descend method. According to numerical solution of
resulting equations, the gravitational clustering can be associated with two
microcanonical phase transitions. A first phase transition with a continuous
character is associated with breakdown of $SO(4)$ symmetry of this model. The
second one is the gravitational collapse, whose continuous or discontinuous
character crucially depends on the regularization of short-range divergence of
gravitation potential. We also derive the thermodynamic limit of this model
system, the astrophysical counterpart of Gibbs-Duhem relation, the order
parameters that characterize its phase transitions and the equation of state.
Other interesting behavior is the existence of states with negative heat
capacities, which appear when the effects of gravitation turn dominant for
energies sufficiently low. Finally, we comment the relevance of some of these
results in the study of astrophysical and cosmological situations. Special
interest deserves the gravitational modification of the equation of state due
to the local inhomogeneities of matter distribution. Although this feature is
systematically neglected in studies about Universe expansion, the same one is
able to mimic an effect that is attributed to the dark energy: a negative
pressure.
