  Long-range interacting systems may exhibit ensemble inequivalence and can
possibly attain equilibrium states under completely open conditions, for which
energy, volume and number of particles simultaneously fluctuate. Here we
consider a modified version of the Thirring model for self-gravitating systems
with attractive and repulsive long-range interactions in which particles are
treated as hard spheres in dimension d=1,2,3. Equilibrium states of the model
are studied under completely open conditions, in the unconstrained ensemble, by
means of both Monte Carlo simulations and analytical methods and are compared
with the corresponding states at fixed number of particles, in the
isothermal-isobaric ensemble. Our theoretical description is performed for an
arbitrary local equation of state, which allows us to examine the system beyond
the excluded-volume approximation. The simulations confirm the theoretical
prediction of the possible occurrence of first-order phase transitions in the
unconstrained ensemble. This work contributes to the understanding of
long-range interacting systems exchanging heat, work and matter with the
environment.
