In statistical physics theory，once ones thought different ensembles to be
equivalent，because equivalence
of ensembles had been proven rigorously in the traditional theory.
Therefore，ones use equivalent thought methods to do some theoretical work，and select the most computationally
convenient ensemble to study the thermodynamic properties. Canonical ensemble is the most convenient for theoretical
calculation，the microcanonical ensemble is the most convenient for numerical calculation，and the grand canonical
ensemble is the most convenient to calculate the thermodynamic properties of quantum systems. Therefore，the
existing new and old textbooks generally only talk about these three ensembles. However，in the 1970s，some
researchers proposed that the canonical ensemble is nonequivalent to the microcanonical ensemble for systems with long-range
interactions in the vicinity of phase transition point. In recent years，some scholars suggested that the two
ensembles are also nonequivalent for systems with short-range interactions in the vicinity of phase transition point，and
more and more research results show that
ensemble is nonequivalent. Therefore，it is imperative to update concepts
and statistical thermodynamics textbooks. In this paper，a simple system with correlative average variables is taken
as an example，which indicates that this "correlation" can also lead to the nonequivalence of the ensemble.
Just Accepted
