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Equilibrium States, Ensembles, and Thermodynamic Limits

Equilibrium field theory connects a probability distribution over fields to measurable response and phase structure. Partition functions organize this connection, but finite-volume fluctuations and infinite-volume phases answer different questions.

Enter this chapter

Section titled “Enter this chapter”

Begin with Statistical Ensembles and Field Configurations to construct a normalized ensemble with specified conserved quantities.

Helpful background. Probability Spaces, Random Variables, and Conditional Expectation and Hamiltonian Initial Data and Phase Space provide probability measures and Hamiltonian phase space.

Topics

Section titled “Topics”
  1. Statistical Ensembles and Field Configurations — Defines configuration spaces, measures, constraints, and microcanonical, canonical, and grand-canonical states.
  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-VV rate functions, constrained branches, interface suppression, and finite-sample bimodality.

How the ideas fit together

Section titled “How the ideas fit together”

A source changes the ensemble, so derivatives of its logarithmic partition function measure connected fluctuations. Integrating a spatial correlation function gives a susceptibility; expanding about a stable saddle gives its Gaussian approximation. This links response, correlation length, and the failure of a mean-field description near criticality Kardar 2007, chs. 1–3.

For a regulated finite system, a two-peaked distribution can indicate coexistence-like behavior without a thermodynamic singularity. Phase selection requires an infinite-volume limit with specified sources or boundary conditions. A probability barrier also supplies no decay time: dynamics must be added separately.

Review the chapter

Section titled “Review the chapter”

For a classical grand-canonical ensemble with charge QQ, why is its susceptibility nonnegative? Hold temperature TT and volume VV fixed.

Solution

Writing β=1/T\beta=1/T and Ξ=∫e−β(H−μQ) dΓ\Xi=\int e^{-\beta(H-\mu Q)}\,\mathrm d\Gamma, differentiation gives ∂μlog⁡Ξ=β⟨Q⟩\partial_\mu\log\Xi=\beta\langle Q\rangle and ∂μ2log⁡Ξ=β2Var⁡(Q)\partial_\mu^2\log\Xi=\beta^2\operatorname{Var}(Q). Thus χ=V−1∂μ⟨Q⟩=βVar⁡(Q)/V≥0\chi=V^{-1}\partial_\mu\langle Q\rangle=\beta\operatorname{Var}(Q)/V\ge0. A negative result signals an inconsistent derivative or an unstable approximate branch.

Where to go next

Section titled “Where to go next”

Continue to KMS States, Imaginary Time, and Thermal Spectra to derive the equilibrium condition from a Gibbs trace.

References

Section titled “References”