Thermodynamic consistency of the equation of state of strongly interacting matter

Addressing strongly interacting matter in the region of energy density where the hadronic gas phase coexists with the quark-gluon plasma phase, we discuss how thermodynamic consistency can be used to constrain the equation of state for uniform matter. We illustrate the method by constructing a $T_c$-dependent family of thermodynamically consistent equations of state based on simple spline interpolations between the gas and plasma phases.

Figure 1

(Color online) The equation of state obtained by interpolating between the hadronic gas and the quark-gluon plasma by means of a cubic spline function (solid). Also shown are the equations of state obtained by augmenting the spline (13) with a quartic adjustment (14) using $p_0=\pm 50\text{MeV}/{\text{fm}}^3$ (dashed).

Figure 2

(Color online) The inverse temperature $\beta \left(\epsilon \right)$ as obtained by applying the thermodynamic integral relation (9) to the functions $p\left(\epsilon \right)$ shown in Fig. 1, starting from ${\epsilon }_H$ and ending at ${\epsilon }_Q$.

Figure 3

(Color online) The equation of state $p\left(\epsilon \right)$ obtained with the cubic spline approximation (13) for three values of the critical temperature, $T_c=160,170,$ and $180\text{MeV}$. The coexistence points are indicated and, for $T_c=170\text{MeV}$, the constant-pressure Maxwell construction is indicated.

Figure 4

(Color online) The dependence of the temperature $T$ on energy density $\epsilon$ as obtained by using Eq. (9) with the three equations of state $p\left(\epsilon \right)$ shown in Fig. 3, The phase-coexistence points are indicated. (All three curves have slight discontinuities at ${\epsilon }_Q$ that are barely visible to the eye.)