The physical idea behind ensemble theory has been discussed in the previous chapter. It is that a real system in thermodynamic equilibrium with fixed mechanical invariants of motion—particle number, energy, linear and angular momentum—randomly chooses its state of motion among all possible states having the same value of those invariants, with equal a priori probabilities. While often sufficient for qualitative discussion, this idea is too limited to be practical, in two related ways. First, the actual system under consideration usually does not have a fixed energy, but rather a fixed temperature. Similarly, instead of the particle number, we may have a fixed value of the chemical potential. Generally, the ensemble construction should be as flexible as thermodynamics itself, with respect to which of a pair of conjugate variables is fixed and which is fluctuating. Second, effective calculation with fixed energy and particle number turns out to be prohibitively difficult. While that is formally just a technical objection, it is practically capable of scuttling the whole program. The success of ensemble theory is due to Gibbs’ circumventing the combinatorial technicalities by a statistical trick which made calculations of realistic thermalized systems feasible, thus avoiding both limitations at once.