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This work is devoted to the thermodynamics of gravitational clustering, acollective phenomenon with a great relevance in the $N$-body cosmologicalproblem. We study a classical self-gravitating gas of identicalnon-relativistic particles defined on the sphere $\mathbb{S}^{3}\subset\mathbb{R}^{4}$ by considering gravitational interaction that corresponds tothis geometric space. The analysis is performed within microcanonicaldescription of an isolated Hamiltonian system by combining continuumapproximation and steepest descend method. According to numerical solution ofresulting equations, the gravitational clustering can be associated with twomicrocanonical phase transitions. A first phase transition with a continuouscharacter is associated with breakdown of $SO(4)$ symmetry of this model. Thesecond one is the gravitational collapse, whose continuous or discontinuouscharacter crucially depends on the regularization of short-range divergence ofgravitation potential. We also derive the thermodynamic limit of this modelsystem, the astrophysical counterpart of Gibbs-Duhem relation, the orderparameters that characterize its phase transitions and the equation of state.Other interesting behavior is the existence of states with negative heatcapacities, which appear when the effects of gravitation turn dominant forenergies sufficiently low. Finally, we comment the relevance of some of theseresults in the study of astrophysical and cosmological situations. Specialinterest deserves the gravitational modification of the equation of state dueto the local inhomogeneities of matter distribution. Although this feature issystematically neglected in studies about Universe expansion, the same one isable to mimic an effect that is attributed to the dark energy: a negativepressure.

Thermodynamics of gravitational clustering phenomena: $N$-body self-gravitating gas on the sphere $\mathbb{S}^{3}\subset\mathbb{R}^{4}$