# Why are microcanonical, canonical and macrocanonical ensembles called that way?

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?

• I suspect it's primarily because humans love to categorized everything. – Carl Witthoft May 11 '18 at 11:55
• Yeah, but what motivates their name? I've came from physics.stackexchange.com/questions/404872/kinds-of-ensemble/… – MatMorPau22 May 11 '18 at 11:57
• Well, micro - [normal] - macro is a pretty straightforward way to "bin" sets of different cardinality. And it sounds more science-y than "teenycanonical," "regularcanonical," and "ubergigundocanonical" :-) – Carl Witthoft May 11 '18 at 12:09

People often expect names and notations to have some deep meanings, or at least a story behind them, but in more cases than not all there is are some loose associations, if that, described as "historical reasons", which is a polite way of saying "just cuz".

Gibbs introduced the names in Elementary Principles in Statistical Mechanics (1902), and he did not exactly give explanations for them:

"When an ensemble of systems is distributed in phase in the manner described, i. e. when the index of probability is a linear function of the energy, we shall say that the ensemble is canonically distributed, and shall call the divisor of the energy ($\Theta$) the modulus of distribution." (p. 33-34)

"Ensemble" just means collection, of moving particles in this case. One can guess from the context that "canonical" might be due to the use of "canonical coordinates" in setting up the distribution. That name is due to Jacobi, see Why are canonical coordinates canonical?, who explained in Note sur l'intégration des équations différentielles de la Dynamique(1837):

"By means of this theorem we find, through the calculation itself, elements whose differential values, in the disturbed motion, take the simple form which they have in the theorem, a form which I designate in my memory under the name of canonical."

Latin canonicus means "according to rule", and "canon" was traditionally used for ecclesiastical laws in medieval Church Latin, seems like Jacobi is talking about a "head canon". Gibbs is slightly more forthcoming with the microcanonical ensemble:

"Thus, the limiting distribution obtained from the part of a canonical ensemble between two limits of energy, when the difference of the 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." (p.115)

"Micro" might refer to the "indefinitely diminished" difference. Gibbs is explicit on the "grand", although who knows why he did not go for "macrocanonical" instead. Maybe it sounded too close to "microcanonical":

"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. The ensembles which we have hitherto discussed are petit ensembles... If an ensemble, whether grand or petit, is identical so far as generic phases are concerned with one canonically distributed, we shall say that its distribution is canonical with respect to generic phases." (p. 189-191)

Gibbs never actually says "grand canonical ensemble", only "grand ensemble canonically distributed", and there is no real reason to why his "petit" did not take.