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 .