# Equivalence of thermodynamic ensembles

It is often argued that thermodynamic ensembles are equivalent in the sense that no matter what ensemble one uses for the calculations, one should end up in the same macroscopic equations of state. This is due to the fact that distributions are sharply peaked around average values and are essentially Dirac delta functions around averages.

Some quantities, however, are related to fluctuations. For example, the specific heat is related to energy fluctuations via the fluctuation-dissipation theorem. I would expect that, in this case, the actual shape of the peaked distribution function matters because that's what describes the fluctuations. How is it then possible that calculating fluctuation related quantities, such as specific heats or magnetic susceptibility, is still ensemble-independent?

The key concept is "thermodynamic limit".

Equivalence of the ensembles is strictly speaking true only after taking thermodynamic limit. For finite systems each ensemble behaves in a different way. It cannot be otherwise: for finite systems the probability distribution in phase space are different. This is evident for the case of fluctuations, but not only for them.

For example, it is clear that the distribution of velocities cannot be the "usual" Maxwell-Boltzmann (M-B) distribution in the microcanonical ensemble (after the whole energy of the system goes as kinetic energy of a single particle, there is no possibility of having more for the exponentially decaying queue of the M-B distribution). And also average quantities in different ensembles are usually equal only within correction which go to zero with the size.

The reason this point goes often unnoticed is that for pedagogical reasons one starts the study of Statistical Mechanics with non-interacting systems. But those systems are very special. What people have to learn soon or later is that any calculation done in any ensemble at a finite size, should extrapolated at the thermodynamic limit in order to eliminate the ensemble dependence.

Unlike means, fluctuations in microcanonical, canonical, and grand-canonical ensembles are different even in the thermodynamic limit. Look at publications of Begun and Gorenstein for details:

https://arxiv.org/pdf/nucl-th/0410044.pdf

https://arxiv.org/pdf/0706.3290.pdf