Specify what ensemble equivalence means

A gas thermodynamic limit sends particle number and volume to infinity at fixed density while holding the interaction law and relevant intensive parameters fixed. Short-range stable interactions make boundary contributions small compared with bulk energy and allow an additive entropy density. Away from singular regimes, a strictly concave smooth entropy selects one stable canonical energy density. A saddle-point evaluation of the state sum then relates the canonical free energy to the corresponding entropy by a Legendre transformation.

This reasoning needs modification at coexistence, where several macroscopic energy regions can contribute, and near critical points, where correlation lengths become large. Long-range nonadditive systems can retain nonconcave microcanonical entropy even in a large-system limit, giving genuine thermodynamic nonequivalence. Size alone is therefore insufficient. One should state the limiting model and observable rather than claim that every large system and every ensemble becomes interchangeable under any interaction or constraint.

In a canonical system with extensive finite heat capacity, energy variance grows in proportion to particle number while mean energy also grows in proportion to number. The standard deviation divided by mean therefore shrinks as inverse square root of size. Energy is still fluctuating in absolute units; its density becomes sharply concentrated. A fixed-energy ensemble instead has no total-energy variance by construction, so equivalence of bulk state functions cannot mean equality of those total-energy distributions.

Local observables and finite subsystems can approach the same predictions when the remainder acts as a large reservoir, with suitable uniqueness and correlation assumptions. Fluctuation magnitudes, global constraints, and rare-event probabilities can still distinguish ensembles. Even a vanishing relative entropy per particle is a weaker notion than equality of every finite-system probability. Stating the level of equivalence keeps an accurate useful theorem from becoming an unrestricted assertion about all measurements.

Consider N binary sites constrained to have exactly K excitations, all compatible configurations equally likely. Every site has excitation probability K divided by N. Observing one excitation slightly reduces the remaining excitation fraction, so different sites are anticorrelated. A canonical model with independent sites can match the same one-site probability but has no such pair correlation. The total excitation count then fluctuates binomially. This difference is exact at finite size and can be calculated without an asymptotic argument.

As N grows at fixed excitation fraction, the pair covariance from one fixed global count falls as inverse N. A small subsystem therefore becomes approximately independent even though the entire total remains exactly fixed. The sampling lab can display concentration of independent site fractions for increasing size. It cannot impose a fixed excitation total or compute hypergeometric correlations, so the constrained comparison must be done with the explicit counting formulas rather than inferred from its binomial bars.