In statistical mechanics, why microcanonical, canonical and macrocanonical ensemble are called that way?

Is there any reason according to the size of the system they can describe properly ( I don't think so) the order they are estudied or something similiar?


1 Answer 1


The terminology is due to Gibbs (see his famous treatise) and he does not give too much motivation for his choice.

Seemingly, he called the canonical ensemble canonical because it is the natural measure on phase space from the point of view of his approach. Indeed, he states

We consider especially ensembles of systems in which the index (or logarithm) of probability of phase is a linear function of the energy. This distribution, on account of its unique importance in the theory of statistical equilibrium, I have ventured to call canonical[...]

The motivation for microcanonical is even less explicit:

We should arrive at the same result, if we should make the density any function of the energy between the limits $\epsilon'$ and $\epsilon"$, and zero outside of those limits. Thus, the limiting distribution obtained from the part of a canonical ensemble between two limits of when the difference of the energy,limiting energies is indefinitely diminished, is independent of the modulus, being determined entirely by the energy, and is identical with the limiting distribution obtained from a uniform density between limits of energy approaching the same value.

We shall call the limiting distribution at which we arrive by this process microcanonical.

He thus sees the microcanonical measure as a canonical measure conditioned on the energy being in the (vanishingly small) interval $[\epsilon',\epsilon'']$. It is possibly because of this further constraint that he used the prefix micro, but this is not very explicit...

The terminology grand canonical seems to refer to the fact that one considers a larger "phase space", in which the number of particles of various species is not fixed. So grand (large) by opposition to petit (small) (with micro then being the smallest...). In his own words:

Instead of considering, as in the preceding chapters, ensembles of systems differing only in phase, we shall now suppose that the systems constituting an ensemble are composed of particles of various kinds, and that they differ not only in phase but also in the numbers of these particles which they contain. The external coordinates of all the systems in the ensemble are supposed, as heretofore, to have the same value, and when they vary, to vary together. For distinction,we may call such an ensemble a grand ensemble, and one in which the systems differ only in phase a petit ensemble. A grand ensemble is therefore composed of a multitude of petit ensembles.

I am well aware that I haven't really answered your question, but since Gibbs introduced this terminology in this book, and does not seem to give much more information on his motivation for the terms used, I doubt that one can do much more than guess. Maybe he's written about that elsewhere (letters?)...

  • $\begingroup$ You might also ask the question on History and Science and Mathematics SE, as somebody might know of other documents where Gibbs might explain his choices. Note that, at least, his terminology is simpler than that by Boltzmann (monode, holode, ergode, etc.). $\endgroup$ Commented May 9, 2018 at 14:22

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