 The last clause is essential because different constraints single out different distributions. A common choice is to keep both the energy and the particle number in the system fixed. One can prove that under these circumstances \(S_G (\varrho)\) is maximal for the so-called microcanonical distribution (or microcanonical ensemble). If we choose to hold the number of particles constant while allowing for energy fluctuations around a given mean value we obtain the so-called canonical distribution; if we also allow the particle number to fluctuate around a given mean value we find the so-called grand-canonical  distribution.[17] 