Thermal (thermodynamic) equilibrium is the macroscopic state a system relaxes to when in prolonged contact with a heat bath: its intensive variables (temperature, pressure, chemical potential) are uniform and time-independent, and there is no net macroscopic flow of energy or matter. Equilibrium is the setting in which the partition function and all of Statistical Mechanics apply.

For a system in equilibrium with a bath at temperature \(T\), the probability of a microstate of energy \(E_n\) is the Boltzmann distribution

\[ P_n = \frac{e^{-\beta E_n}}{Z},\qquad Z = \sum_n e^{-\beta E_n}=\mathrm{Tr}\,e^{-\beta H},\qquad \beta=\frac{1}{k_BT}. \]

Equilibrium expectation values are then

\[ \langle \mathcal O\rangle = \frac{1}{Z}\,\mathrm{Tr}\big[\mathcal O\,e^{-\beta H}\big]. \]

Equilibrium is characterized by maximum entropy consistent with the conserved constraints, or equivalently minimum free energy \(F=-T\ln Z\).

For quantum fields at temperature \(T\), computing equilibrium properties means evaluating

\[ \langle \mathcal O\rangle = \frac{1}{Z}\int [D\phi]\ \mathcal O[\phi]\ e^{-S_E[\phi]}, \]

a Euclidean path integral with compact time \(\tau\in[0,\beta]\). This is the framework in which one studies equilibrium QCD — the equation of state, screening lengths, and the deconfinement transition — and it is exactly what lattice Monte Carlo simulations sample. (Equilibrium says nothing about transport or real-time dynamics, which require analytic continuation or a Minkowski formulation.)