Video Upload Options

References

1. Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics. New York: Charles Scribner's Sons. 
2. Simulation of chemical reaction equilibria by the reaction ensemble Monte Carlo method: a review https://doi.org/10.1080/08927020801986564
3. This equal-volume partitioning is a consequence of Liouville's theorem, i. e., the principle of conservation of extension in canonical phase space for Hamiltonian mechanics. This can also be demonstrated starting with the conception of an ensemble as a multitude of systems. See Gibbs' Elementary Principles, Chapter I.
4. (Historical note) Gibbs' original ensemble effectively set h = 1 [energy unit]×[time unit], leading to unit-dependence in the values of some thermodynamic quantities like entropy and chemical potential. Since the advent of quantum mechanics, h is often taken to be equal to Planck's constant in order to obtain a semiclassical correspondence with quantum mechanics.
5. In some cases the overcounting error is benign. An example is the choice of coordinate system used for representing orientations of three-dimensional objects. A simple encoding is the 3-sphere (e. g., unit quaternions) which is a double cover—each physical orientation can be encoded in two ways. If this encoding is used without correcting the overcounting, then the entropy will be higher by k log 2 per rotatable object and the chemical potential lower by kT log 2. This does not actually lead to any observable error since it only causes unobservable offsets.
6. Technically, there are some phases where the permutation of particles does not even yield a distinct specific phase: for example, two similar particles can share the exact same trajectory, internal state, etc.. However, in classical mechanics these phases only make up an infinitesimal fraction of the phase space (they have measure zero) and so they do not contribute to any volume integral in phase space.