Newtonian gravity in d dimensions

We study the influence of the dimension of space on the thermodynamics of the classical and quantum self-gravitating gas. We consider Hamiltonian systems of self-gravitating particles described by the microcanonical ensemble and self-gravitating Brownian particles described by the canonical ensemble. We present a gallery of caloric curves in different dimensions of space and discuss the nature of phase transitions as a function of the dimension d. We also provide the general form of the Virial theorem in d dimensions and discuss the particularity of the dimension d = 4 for Hamiltonian systems and the dimension d = 2 for Brownian systems. To cite this article: P.-H. Chavanis, C. R. Physique 7 (2006).