Thermodynamic Limits, Correlation Decay, and Phase Control

A thermodynamic limit asks whether expectations of local observables stabilize as the containing region fills space. Compactness can produce subsequences; decay or monotonicity controls boundaries; uniqueness determines whether every exhaustion and boundary condition gives the same state. At coexistence, several legitimate infinite-volume phases may survive.

Required background. Cluster expansions and correlation inequalities supplies uniform connected bounds; interacting measures, stability, and Wick ordering supplies the finite-volume measures.

Helpful background. Clustering, vacuum uniqueness, and mass-gap implications interprets decay; factorial KMS states, phases, and symmetry breaking distinguishes pure phases; thermodynamic limits, phases, and ensemble equivalence supplies the statistical-mechanical setting.

Local convergence and boundary control

Let $\mu_{\Lambda}^{b}$ be a finite-volume measure with boundary condition $b$, and let $F$ depend only on fields in a compact set $K$. A strong uniqueness estimate has the form

$$|\mathbb E_{\mu_\Lambda^b}F-\mathbb E_{\mu_\Lambda^{b'}}F| \le C_F e^{-c\,\mathrm{dist}(K,\partial\Lambda)}$$

uniformly over admissible $b,b'$. Together with consistency estimates between nested volumes, it makes $\mathbb E_{\mu_\Lambda^b}F$ Cauchy and independent of boundary conditions. If cylinder observables form a determining class and their moments obey uniform growth bounds, the limiting functional defines a probability measure.

A weaker route uses tightness. Uniform local moments yield subsequential weak limits in a distribution topology. Translation invariance, reflection positivity, and correlation inequalities pass to the limit when their test functionals are continuous and uniformly integrable. But tightness alone does not identify the phase.

A massive three-dimensional scalar example

Consider a lattice-regulated $\phi^4_3$ model in periodic boxes $\Lambda_L$ in a massive weak-coupling regime. After the required local mass and vacuum counterterms have been fixed, a convergent cluster or renormalization-group analysis provides uniform bounds for local connected correlations. For a local polynomial observable $F$, polymers that touch both its support and the distant boundary carry an exponentially small weight. Therefore

$$\lim_{L\to\infty}\mathbb E_{\mu_{Lambda_L}^{\mathrm{per}}}F$$

exists, and the same limit holds for boundary conditions within the theorem’s controlled class. This is a statement about local correlation functions, not convergence in total variation of probability measures on an ever-growing configuration space. The role of lattice geometry and boundary conditions is developed in lattice geometry, boundaries, and anisotropy.

For the continuum $\phi^4_3$ construction, simultaneous ultraviolet and volume removal of Schwinger functions and the OS conclusion were proved using phase-cell/cluster methods and other techniques; Summers 2016, §3.2, pp. 16–17 states the precise low-dimensional status. The thermodynamic conclusion here does not itself remove the lattice spacing: volume and ultraviolet limits remain separate obligations.

An independent check uses covariance. At zero interaction, the difference between periodic finite-volume and infinite-volume massive Green functions on a fixed compact set is a sum over nonzero image points. The mass makes that sum exponentially small in $L$. The interacting boundary estimate should reduce continuously to this image bound as the coupling tends to zero.

Coexistence is not a failed limit

For a double-well $P(\phi)_2$ interaction at low temperature and zero external field, plus and minus boundary conditions can converge to distinct infinite-volume measures $\mu_+$ and $\mu_-$. Their order parameters satisfy $\mathbb E_{\mu_+}\phi(f)>0$ and $\mathbb E_{\mu_-}\phi(f)<0$ for positive $f$. Both are valid phases and can each cluster. Summers 2016, §3.1, pp. 13–14 describes this constructive phase-coexistence result.

Thus “the thermodynamic limit exists” must say whether it means existence along a specified boundary condition, uniqueness across boundaries, or convergence to a mixture. Exponential clustering in a pure phase does not imply uniqueness of the global Gibbs state; a symmetric mixture of two clustering phases generally fails clustering.

Adversarial phase test

Assume boundary independence while taking the low-temperature double-well model at zero external field. The plus and minus one-point functions have opposite signs, so the proposed common limit cannot exist. What survives is subsequential or boundary-specific convergence and a phase decomposition.

A second failure sends $L\to\infty$ while allowing the mass parameter to approach zero without a uniform decay rate. The boundary estimate degenerates because $c\downarrow0$. Fixed-mass thermodynamic control does not license the critical limit.

Thus every volume-limit statement should record the exhaustion and allowed boundary class, even when the final state is translation invariant.

Exercises

1. Mixtures and clustering. Let $\mu=\tfrac12(\mu_++\mu_-)$ with order parameters $\pm v$. Show why $\mu$ does not cluster for the order parameter.

Solution

The mixture has one-point function zero, while the far-separated two-point function tends to $\tfrac12v^2+\tfrac12v^2=v^2$. Its truncated two-point function therefore does not tend to zero.

2. Cauchy estimate. If nested boxes obey $|E_LF-E_{L'}F|\le 2C_Fe^{-cL}$ for $L'\ge2L$, explain why a dyadic sequence converges.

Solution

Along $L_n=2^nL_0$, the telescoping tail is bounded by $2C_F\sum_{n\ge N}e^{-cL_n}$, which tends to zero. Hence the expectations are Cauchy.

References

• Feldman, Joel, and Konrad Osterwalder. “The Wightman Axioms and the Mass Gap for Weakly Coupled $\phi^4_3$ Quantum Field Theories.” Annals of Physics 97 (1976): 80–135. DOI.
• Glimm, James, Arthur Jaffe, and Thomas Spencer. “Phase Transitions for $\phi^4_2$ Quantum Fields.” Communications in Mathematical Physics 45 (1975): 203–216. DOI.
• Summers, Stephen J. “A Perspective on Constructive Quantum Field Theory.” arXiv:1203.3991, revised 2016. Open PDF.