Equivalence of microcanonical and canonical ensembles (thermodynamic limit)

Under concavity/regularity of the entropy and short-range interactions, microcanonical and canonical ensembles yield the same macroscopic predictions in the thermodynamic limit.

Equivalence of microcanonical and canonical ensembles (thermodynamic limit)

Statement (ensemble equivalence)

Consider a sequence of $N$ -particle (or finite-volume) systems with Hamiltonian (see Hamiltonian or lattice Hamiltonian ). Let

$\mu^{\text{mc}}_{N,u}$ be the microcanonical measure at energy density $u$ ,
$\mu^{\text{can}}_{N,\beta}$ be the canonical ensemble at inverse temperature $\beta$ .

Assume a thermodynamic entropy density $s(u)$ exists and is concave on the relevant energy range, and that the canonical free-energy potential $\psi(\beta)$ exists and is related to $s$ by Legendre duality (see Legendre duality between free energy and entropy ).

Then for any “macroscopic” observable $A_N$ whose fluctuations are controlled by the energy (e.g. any local observable in a short-range lattice system), if $u=u(\beta)$ is the equilibrium energy density at $\beta$ , one has

\lim_{N\to\infty}\Big(\mathbb E_{\mu^{\text{mc}}_{N,u(\beta)}}[A_N]-\mathbb E_{\mu^{\text{can}}_{N,\beta}}[A_N]\Big)=0,

and both ensembles concentrate on the same set of equilibrium macrostates.

If $s(u)$ is not concave (e.g. due to phase coexistence or long-range interactions), then the ensembles can be nonequivalent: microcanonical equilibrium at a given $u$ need not correspond to any canonical equilibrium at any $\beta$ .

Key hypotheses

Existence of thermodynamic limits for entropy/free energy (often from subadditivity methods).
Short-range (or otherwise “well-behaved”) interactions ensuring additivity and concentration.
Concavity (stability) of the entropy density $s(u)$ on the relevant domain.
A large-deviation description of the energy or macrovariables (see large deviation principle ).

Conclusions

Same thermodynamics: equations of state computed from microcanonical or canonical descriptions agree in the thermodynamic limit.
Same typical macrostates: both ensembles concentrate on the same equilibrium set when matched by $u(\beta)$ .
Inequivalence criterion: nonconcavity of $s(u)$ signals potential ensemble inequivalence and can correspond to phase transitions.

Cross-links to definitions

Ensembles and observables: microcanonical measure , canonical ensemble , ensemble average .
Entropy/free energy: Boltzmann entropy , statistical free energy , entropy–free energy duality .
Large deviations: LDP , large-deviation characterization of equilibrium .

Proof idea / significance

The canonical ensemble is a mixture over energies; when the energy satisfies a large-deviation principle with strictly concave entropy, the mixture concentrates exponentially near the unique maximizer of $s(u)-\beta u$ . This concentration forces expectations of suitably regular observables to match those under the microcanonical ensemble at the corresponding energy. The concavity requirement is exactly what guarantees that the Legendre transform correctly inverts and that no “hidden” nonconcave branches appear.