Statistical mechanics relates microstates to observables via ensembles such as microcanonical, canonical, and grand canonical. The partition function $Z$ is the central generating object, with thermodynamic potentials obtained from logarithms and derivatives of $Z$. Fluctuations become especially important near phase transitions and in finite systems.

A phase transition often involves changes in an order parameter and the emergence or loss of symmetry. First-order transitions exhibit latent heat, while continuous transitions show diverging correlation length and power-law scaling. Renormalization-group ideas classify critical points by universality.

Two-point correlation functions probe coherence and collective behavior. Near criticality, the correlation length $\xi$ can diverge, and observables scale as powers characterized by critical exponents. This is where field-theoretic methods become especially powerful.

Entropy and entanglement have become central in modern condensed matter, quantum information, and black hole physics. Entanglement entropy can diagnose topological phases, criticality, and hidden structure not visible in local order parameters alone.