The microcanonical statistical mechanics of a set of self-gravitating particles is analyzed in a mean-field approach. In order to deal with an upper bounded entropy functional, a softened gravitational potential is used. The softening is achieved by truncating to N terms an expansion of the Newtonian potential in spherical Bessel functions. The order N is related to the softening at short distances. This regularization has the remarkable property that it allows for an exact solution of the mean-field equation. It is found that for N not too large the absolute maximum of the entropy coincides to high accuracy with the solution of the Lane-Emden equation, which determines the mean-field mass distribution for the Newtonian potential for energies larger than $E_c\approx -{0.335GM}^2/R.\mathrm{}$ Below this energy a collapsing phase transition, with negative specific heat, takes place. The dependence of this result on the regularizing parameter N is discussed.

• Received 13 December 1999

DOI:https://doi.org/10.1103/PhysRevE.61.6270