Statistical mechanics is the framework that derives the macroscopic thermodynamic behaviour of a many-body system from the statistics of its microscopic states. Its central object is the partition function, and it is the bridge that turns quantum field theory at finite temperature into a computable Euclidean problem.
Depending on which quantities are held fixed one uses a different ensemble:
- Microcanonical — fixed energy \(E\); entropy \(S=k_B\ln\Omega(E)\).
- Canonical — fixed temperature \(T\); the workhorse for field theory, \[ Z = \mathrm{Tr}\,e^{-\beta H},\qquad \beta=\frac{1}{k_BT}. \]
- Grand canonical — fixed \(T\) and chemical potential \(\mu\); \(Z=\mathrm{Tr}\,e^{-\beta(H-\mu N)}\), relevant for QCD at finite baryon density.
Once \(Z\) is known, all equilibrium thermodynamics follows by differentiation:
The formal identity \(e^{-\beta H}\leftrightarrow\) imaginary-time evolution unifies statistical mechanics with the path integral formulation of QFT: a quantum field theory in \(d\) spatial dimensions at temperature \(T\) is a classical statistical system in \(d+1\) dimensions with one compact (euclidean) time direction of length \(\beta\). This equivalence is the conceptual foundation of lattice QCD — the equilibrium properties of quarks and gluons are extracted by sampling field configurations with statistical-mechanical Monte Carlo methods, weighting each by \(e^{-S_E}\).