Modified Thirring model beyond the excluded-volume approximation

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