Long-range interacting systems in the unconstrained ensemble

Completely open systems can exchange heat, work, and matter with the environment. While energy, volume, and number of particles fluctuate under completely open conditions, the equilibrium states of the system, if they exist, can be specified using the temperature, pressure, and chemical potential as control parameters. The unconstrained ensemble is the statistical ensemble describing completely open systems and the replica energy is the appropriate free energy for these control parameters from which the thermodynamics must be derived. It turns out that macroscopic systems with short-range interactions cannot attain equilibrium configurations in the unconstrained ensemble, since temperature, pressure, and chemical potential cannot be taken as a set of independent variables in this case. In contrast, we show that systems with long-range interactions can reach states of thermodynamic equilibrium in the unconstrained ensemble. To illustrate this fact, we consider a modification of the Thirring model and compare the unconstrained ensemble with the canonical and grand-canonical ones: The more the ensemble is constrained by fixing the volume or number of particles, the larger the space of parameters defining the equilibrium configurations.