Equilibrium States, Ensembles, and Thermodynamic Limits

Equilibrium field theory begins with a measure, its controlled variables, and its limits. Partition functions generate static response; connected correlations determine susceptibilities and length scales; Gaussian fluctuations diagnose local stability; thermodynamic limits create genuine phases; and large deviations quantify rare macrostates. The organizing rule is simple: a finite-volume identity, an infinite-volume phase statement, and a dynamical transition claim are different levels of conclusion.

The field-theoretic equilibrium route used here is developed systematically in Kardar 2007, chs. 1–3.

Enter this chapter

You are ready to begin if you can normalize a probability measure and identify the Hamiltonian and conserved data of a regulated field system. If either step is uncertain, review Probability Spaces, Random Variables, and Conditional Expectation and Hamiltonian Initial Data and Phase Space.

Three quick diagnostics locate the best entry.

• Can you say which variables are held fixed in a derivative of $\log Z$? If yes, enter at partition functions; otherwise start with ensembles.
• Can you distinguish a screening length from a real-time damping time? If yes, enter at correlations; otherwise use the Euclidean–classical correspondence first.
• Can you explain why $\langle M\rangle=0$ at finite volume in a symmetric system with a two-peaked histogram? If yes, enter at thermodynamic limits; otherwise follow the core route in order.

Choose a route

Goal	Suggested route	Capability gained
First thermal-field-theory encounter	Ensembles → partition functions → correlations → limits and phases	Move from a normalized state to response and a valid phase statement
Critical phenomena	Euclidean fields → correlations → Gaussian fluctuations → limits	Diagnose scale growth and the breakdown of mean-field control
First-order coexistence	Partition functions → limits and phases → large deviations	Separate bulk, interface, constrained, and finite-size evidence
Euclidean data interpretation	Euclidean fields → correlations → Chapter 2	Know which static quantities continue and which require spectral input

These are reading routes, not additional prerequisite edges. Each page states its own required background.

The chapter’s logic

The regulated ensemble is the starting object. Adding sources produces $\log Z$, whose derivatives are cumulants and response coefficients. Spatially resolved cumulants give structure factors and correlation lengths. Expanding the free energy about a stationary configuration gives a Hessian, which controls the Gaussian covariance only while its nonzero spectrum is positive and interactions among fluctuations are small.

The infinite-volume step changes the ontology. Finite-volume analyticity allows sharp crossovers and long-lived sectors but not a strict singular phase transition. A pure phase requires an order of limits or a selecting boundary condition. Large-deviation functions then organize rare macrostates, while interface costs can occur at a lower scaling speed than the bulk volume.

The same chain prevents three category errors:

1. a configuration is not an ensemble;
2. a nonconvex coarse-grained landscape is not the exact convex thermodynamic potential; and
3. a static probability barrier is not a nucleation rate.

Guide to the seven pages

1. Statistical Ensembles and Field Configurations defines configuration spaces, measures, constraints, and microcanonical, canonical, and grand-canonical states. It contains the shared convention-and-limit table.
2. Partition Functions and Thermodynamic Response derives potentials, expectation values, connected cumulants, susceptibilities, and Legendre transforms from one source-dependent partition function.
3. Euclidean Fields and Classical Statistical Systems identifies the exact regulated statistical correspondence and the additional positivity hypotheses required for Lorentzian reconstruction.
4. Correlations, Susceptibilities, and Correlation Lengths relates integrated connected correlations, structure factors, screening poles, and finite-volume length estimators.
5. Gaussian Fluctuations and Effective Free Energy treats Hessians, determinants, zero and negative modes, and the Ginzburg breakdown test.
6. Thermodynamic Limits, Phases, and Ensemble Equivalence explains phase selection, clustering, convex duality, and the conditions for ensemble equivalence.
7. Large Deviations and Phase Coexistence distinguishes speed-$V$ rate functions, constrained branches, interface suppression, and finite-sample bimodality.

Conventions and recurring example

The chapter inherits the site’s global conventions. It uses $\beta=1/T$, and every ensemble formula states whether its exponent is a Hamiltonian divided by $T$ or an already dimensionless Euclidean action. Volume, regulator, boundary conditions, reference measure, source sign, and order of limits remain local data.

A regulated scalar field supplies the recurring example. Its canonical measure introduces ensembles; source derivatives give the equation of state and susceptibility; its Gaussian covariance yields a screening length; its Hessian reveals critical enhancement; a selecting source demonstrates noncommuting limits; and a constrained order-parameter distribution distinguishes bulk and interface large deviations. The example does not stand in for gauge constraints, long-range forces, or nonpositive weights.

Review the chapter

Retrieval. State the difference among a microstate, macrostate, and ensemble. A correct answer identifies a point in configuration space, constrained extensive data, and a normalized measure.

Derivation. Starting from $\Xi(\mu)$, derive $\chi=\beta\langle(\Delta Q)^2\rangle/V$. A correct answer holds $T,V$ fixed and differentiates $\log\Xi$, not $\Xi$.

Representation change. Translate a Euclidean scalar covariance into a classical structure factor. A correct answer matches the dimensionless weight and refuses to infer a real-time damping rate.

Failure diagnosis. A finite simulation shows two peaks and a negative fitted quartic potential. Explain why this is insufficient evidence for two thermodynamic phases. A correct answer asks for volume scaling, phase selection, interface cost, sampling overlap, and the status of the approximate potential.

Synthesis. Describe the minimal chain from a regulated Hamiltonian to a phase-coexistence claim. A correct answer includes normalization, source response, correlation-scale control, $V\to\infty$, source or boundary limit, and a declared large-deviation speed.

Continue

KMS States, Imaginary Time, and Thermal Spectra replaces finite-volume statistical language by the intrinsic equilibrium condition and connects it to Euclidean and real-time correlators. Critical and Stochastic Dynamics adds dynamics to equilibrium free energies. Thermal Phases, Metastability, and Nucleation develops decay rates only after the static phase and metastability questions have been settled.

References

• Ellis, Richard S. Entropy, Large Deviations, and Statistical Mechanics. New York: Springer, 1985. doi:10.1007/978-1-4613-8533-2.
• Kardar, Mehran. Statistical Physics of Fields. Cambridge: Cambridge University Press, 2007. doi:10.1017/CBO9780511815881.
• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. 5th ed. Oxford: Oxford University Press, 2021. doi:10.1093/oso/9780198834625.001.0001.