QCD thermodynamics from lattice calculations with nonequilibrium methods: The SU(3) equation of state

A precise lattice determination of the equation of state in SU(3) Yang-Mills theory is carried out by means of a simulation algorithm, based on Jarzynski’s theorem, that allows one to compute physical quantities in thermodynamic equilibrium, by driving the field configurations of the system out of equilibrium. The physical results and the computational efficiency of the algorithm are compared with other state-of-the-art lattice calculations, and the extension to full QCD with dynamical fermions and to other observables is discussed.

Figure 1

Results for $p/T^4$, as a function of $T/T_c$, from simulations at different values of $N_t$. The inset shows a zoom onto the confining phase, $T<T_c$.

Figure 2

Our results for $p/T^4$, extrapolated to the continuum (green circles), as a function of $T/T_c$, in comparison with those obtained with the integral method in Ref. [17] (red squares) and with those obtained using the moving-frame method in Ref. [12] (blue triangles). The results in the confining phase are displayed in the inset plot.

Figure 3

Our continuum-extrapolated results for $\mathrm{\Delta }/T^4$ (green circles), as a function of $T/T_c$, in comparison with those obtained with the integral method in Ref. [17] (red squares), with those obtained using the moving-frame method in Ref. [12] (blue triangles), and with those computed using the gradient-flow method in Ref. [9] (orange diamonds).

Figure 4

Same as in Fig. 2, but for the energy density in units of the fourth power of the temperature.

Figure 5

Same as in Fig. 3, but for the entropy density in units of the third power of the temperature.

Figure 6

Histograms of the probability density $p(\mathrm{\Delta }S)$ for a variation $\mathrm{\Delta }S$ in Euclidean action during nonequilibrium trajectories, as defined in Eq. (29), in two different calculations at finite temperature, on a lattice with $N_t=6$ and $N_s=96$. In both simulations, the starting configuration in each of the trajectories is drawn from an equilibrated distribution at $\beta ={\beta }_{\mathrm{in}}=6.17921$, and is let evolve through a sequence of nonequilibrium transformations, during which the $\beta$ parameter is decreased down to ${\beta }_{\mathrm{fin}}=6.13671$. In the first simulation (red histogram), the nonequilibrium transformations were carried out by dividing the $({\beta }_{\mathrm{fin}}-{\beta }_{\mathrm{in}})$ interval into $N=425$ subintervals of amplitude $|\mathrm{\Delta }\beta |=|-{10}^{-4}|$ and 300 trajectories were produced, whereas in the second simulation (green histogram), the $({\beta }_{\mathrm{fin}}-{\beta }_{\mathrm{in}})$ interval was divided into $N=4250$ subintervals of amplitude $|\mathrm{\Delta }\beta |=|-{10}^{-5}|$, and the number of trajectories was $n_{\mathrm{traj}}=40$. The estimates for the free-energy difference (in units of the temperature) obtained using the algorithm based on Jarzynski’s equality from these two simulations are $\mathrm{\Delta }F/T=825741.5\pm 4.1$ for the calculation with $N=425$, and $\mathrm{\Delta }F/T=825740.3\pm 0.5$ for the one with $N=4250$.

Figure 7

Same as in Fig. 6, but for transformations in which $\beta$ is let vary from ${\beta }_{\mathrm{in}}=6.13671$ to ${\beta }_{\mathrm{fin}}=6.17921$. The red histogram shows the distribution of $\mathrm{\Delta }S$ values obtained in simulations with $N=425$ and $\mathrm{\Delta }\beta ={10}^{-4}$ (with 300 trajectories), whereas the green histogram displays the probability density for the $\mathrm{\Delta }S$ values obtained in simulations with $N=4250$ and $\mathrm{\Delta }\beta ={10}^{-5}$, for which 40 trajectories were produced. The results for the free-energy difference (in units of the temperature) from these two simulations are $\mathrm{\Delta }F/T=-825734.7\pm 2.6$, for the calculation with $N=425$, and $\mathrm{\Delta }F/T=-825740.3\pm 1.1$, for the one with $N=4250$.