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.

Figure 1

Reduced number of particles $\overline{x}$ for the modified Thirring model with $b=-1$ in the unconstrained ensemble: in (a), as a function of the reduced pressure $p$ with constant fugacity $z$, and, in (b), as a function of the fugacity with constant reduced pressure. We note that in this and in the following four figures we have, as requested by the equilibrium condition in the unconstrained ensemble, $p<z$ and $z<1/\left(2e\right)$.

Figure 2

Reduced density $\overline{y}$ for the modified Thirring model with $b=-1$ in the unconstrained ensemble: in (a), as a function of the reduced pressure $p$ with constant fugacity $z$, and, in (b), as a function of the fugacity with constant reduced pressure.

Figure 3

Reduced volume $\overline{v}$ for the modified Thirring model with $b=-1$ in the unconstrained ensemble: in (a), as a function of the reduced pressure $p$ with constant fugacity $z$, and, in (b), as a function of the fugacity with constant reduced pressure.

Figure 4

(a) Reduced volume $\overline{v}$ as a function of the reduced pressure $p$ in the unconstrained ensemble, where $\overline{x}$ is held constant and $b=-1$. (b) Reduced volume as a function of the reduced pressure at constant $\overline{x}$ but with $b=-0.1$.

Figure 5

(a) Reduced density $\overline{y}$ as a function of the fugacity $z$ in the unconstrained ensemble, where the reduced volume $\overline{v}$ is held constant and $b=-1$. (b) Reduced density as a function of the fugacity at constant $\overline{v}$ but with $b=-0.1$.

Figure 6

Modified Thirring model in the grand-canonical ensemble. (a) Reduced pressure as a function of the fugacity for constant reduced volume $v$. For $b<0$, the reduced pressure is always smaller than the fugacity, while, for $b>0$, it is shown that $p>z$. In the case $b=0$, the condition $p=z$ is always satisfied. (b) Pressure as a function of the volume with constant reduced number of particles $\overline{x}$. In the case $b=0.5$, the portion of the curve with positive slope correspond to states of negative isothermal compressibility.

Figure 7

Comparison between the unconstrained and the canonical ensembles in the modified Thirring model. (a) The black lines indicate the reduced pressure $p$ as a function of the reduced volume $\overline{v}$ in the unconstrained ensemble, where the reduced number of particles $\overline{x}$ is held constant for $b=-1$. The red lines show the analogous curves in the canonical ensemble, where $v$ and $x$ are fixed to the same values of $\overline{v}$ and $\overline{x}$, respectively. (b) The plot shows the same as in (a) but with $b=-0.1$ and different values of $\overline{x}$. For $b=-0.1$, the model shows a first-order phase transition in the canonical ensemble, as can be appreciated in the plot from the jumps in pressure (red dotted lines). In both (a) and (b), the curves in the canonical ensemble continue to the right superposed on the curves in the unconstrained ensemble. We remark that in (b), the three curves in the canonical ensemble present a portion with positive slope, indicating negative compressibility. In particular, the slopes of the curves with $\overline{x}=2.4$ and $\overline{x}=2.9$ are positive before the jump.

Figure 8

Comparison between the unconstrained and the canonical ensembles in the modified Thirring model. (a) The black lines represent the fugacity $z$ as a function of the reduced density $\overline{y}$ in the unconstrained ensemble. Here the reduced volume $\overline{v}$ is held constant and we set $b=-1$. The red lines show the analogous curves in the canonical ensemble, where $y$ and $v$ are fixed to the same values of $\overline{y}$ and $\overline{v}$, respectively. In (b), the plot shows the same as in (a) but with $b=-0.1$. The jumps correspond to first-order phase transitions in the canonical ensemble (red dotted lines). In both (a) and (b), the curves in the canonical ensemble continue to the left superposed on the curves in the unconstrained ensemble.