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Ensemble equivalence

TL;DR

The grand-canonical (GC), canonical, and microcanonical (MC) ensembles are different probability laws. They are connected by exact conditioning identities, and their differences may become negligible for some observables in some asymptotic regimes — but that is an empirical and asymptotic question, not an exact equality. Support observables (binary ones) tend to remain sensitive to the ensemble choice.

Exact conditioning identity

Let the grand-canonical law be

\[ P_{\mathrm{GC}}(t)\propto d_F(t)e^{-\theta\cdot C(t)}. \]

On the fiber \(C(t)=C^\star\), the Boltzmann factor

\[ e^{-\theta\cdot C(t)}=e^{-\theta\cdot C^\star} \]

is constant, so the conditional law is

\[ P_{\mathrm{GC}}(t\mid C(t)=C^\star) \propto d_F(t)\mathbf 1[C(t)=C^\star]. \]

This is exact: conditioning the GC law on the hard constraints yields the microcanonical target. It is a conditional identity; it does not imply that unconditional GC and MC are equal.

Canonical conditioning

Where the canonical ensemble exists (ME + STRENGTH),

\[ P_{\mathrm{CAN}}(t)=P_{\mathrm{GC}}(t\mid T=T^\star). \]

Again, this is an exact conditioning identity; the practical differences between conditional and unconditional ensembles are a separate question.

Asymptotic / large-occupation discussion

Differences between conditional and unconditioned ensembles can become negligible for observables insensitive to residual fluctuations of the conditioned statistics, in an appropriate asymptotic regime. They do not "vanish" in any unconditional sense.

The right way to study the question is empirical, at finite \(N\), with the planned GC-vs-MC comparison notebook as the evidence source (Benchmarks and dedicated notebook docs/examples/grand-vs-micro-practical.ipynb).

Support observables

Support uses the discontinuous-at-zero transformation

\[ a_{ij}=\mathbf 1[t_{ij}>0]. \]

Quantities built from it — \(E\), degree sequences \(k\), binary clustering, support motifs — can remain sensitive to the ensemble choice even where smooth occupation observables converge. Support observables cannot therefore be broadly classified as convergent.

Event-family dependence

ME asymptotics do not generalize to B or W without proof. The three families have different pair laws and domains (in particular W has \(q_{ij}\in(0,1)\)), so asymptotic statements must be family-specific. Three kinds of statements need to be told apart in the discussion:

  • exact conditional identities;
  • theoretical asymptotic statements;
  • empirical observations.
flowchart TD
    GC[Grand canonical P_GC] -->|Condition on T=T*| CAN[Canonical]
    GC -->|Condition on hard constraints C=C*| MC[Microcanonical target]

    GC --> F1[Soft fluctuations remain]
    CAN --> F2[T fixed exactly]
    MC --> F3[Hard fiber fixed exactly]

    F1 --> ASY{Asymptotic / empirical comparison}
    F2 --> ASY
    F3 --> ASY

    ASY --> O1[Some smooth occupation observables may approach]
    ASY --> O2[Support observables may remain different]

Conditioning arrows are exact probability identities where the corresponding model exists. The bottom comparison is an asymptotic/empirical question, not an exact equality.

Practical notes

  • A single benchmark or notebook run is evidence, not a theorem.
  • Report conclusions with at least: family, constraint, \(N\), sparsity regime, and observable (see Benchmarks conventions).
  • For the scientific choice between GC and MC, see Choose a model.