Thermodynamics of SU(3) Gauge Theory in 2 + 1 Dimensions

The pressure, and the energy and entropy densities are determined for the SU(3) gauge theory in $2 + 1$ dimensions from lattice Monte Carlo calculations in the interval $0.6 \leq T/T_c \leq 15$. The finite temperature lattices simulated have temporal extent $N_\tau = 2, 4, 6$ and 8, and spatial volumes $N_S^2$ such that the aspect ratio is $N_S/N_\tau = 8$. To obtain the thermodynamical quantities, we calculate the averages of the temporal plaquettes $P_\tau$ and the spatial plaquettes $P_S$ on these lattices. We also need the zero temperature averages of the plaquettes $P_0$, calculated on symmetric lattices with $N_\tau = N_S$. We discuss in detail the finite size ($N_S$-dependent) effects. These disappear exponentially. For the zero temperature lattices we find that the coefficient of $N_S$ in the exponent is of the order of the glueball mass. On the finite temperature lattices it lies between the two lowest screening masses. For the aspect ratio equal to eight, the systematic errors coming from the finite size effects are much smaller than our statistical errors. We argue that in the continuum limit, at high enough temperature, the pressure can be parametrized by the very simple formula $p=a-bT_c/T$ where $a$ and $b$ are two constants. Using the thermodynamical identities for a large homogeneous system, this parametrization then determines the other thermodynamical variables in the same temperature range.