Editing Ensemble (mathematical physics) (section)

==== Correcting overcounting in phase space ====

Typically, the phase space contains duplicates of the same physical state in multiple distinct locations. This is a consequence of the way that a physical state is encoded into mathematical coordinates; the simplest choice of coordinate system often allows a state to be encoded in multiple ways. An example of this is a gas of identical particles whose state is written in terms of the particles' individual positions and momenta: when two particles are exchanged, the resulting point in phase space is different, and yet it corresponds to an identical physical state of the system. It is important in statistical mechanics (a theory about physical states) to recognize that the phase space is just a mathematical construction, and to not naively overcount actual physical states when integrating over phase space. Overcounting can cause serious problems:
* Dependence of derived quantities (such as entropy and chemical potential) on the choice of coordinate system, since one coordinate system might show more or less overcounting than another.<ref group=note>In some cases the overcounting error is benign. An example is the [[Charts on SO(3)|choice of coordinate system used for representing orientations of three-dimensional objects]]. A simple encoding is the [[3-sphere]] (e. g., unit [[quaternion]]s) which is a [[double covering group|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 {{math|''k'' log 2}} per rotatable object and the chemical potential lower by {{math|''kT'' log 2}}. This does not actually lead to any observable error since it only causes unobservable offsets.</ref>
* Erroneous conclusions that are inconsistent with physical experience, as in the [[mixing paradox]].<ref name="gibbs"/>
* Foundational issues in defining the [[chemical potential]] and the [[grand canonical ensemble]].<ref name="gibbs"/>
It is in general difficult to find a coordinate system that uniquely encodes each physical state. As a result, it is usually necessary to use a coordinate system with multiple copies of each state, and then to recognize and remove the overcounting.

A crude way to remove the overcounting would be to manually define a subregion of phase space that includes each physical state only once and then exclude all other parts of phase space. In a gas, for example, one could include only those phases where the particles' {{math|''x''}} coordinates are sorted in ascending order. While this would solve the problem, the resulting integral over phase space would be tedious to perform due to its unusual boundary shape. (In this case, the factor {{math|''C''}} introduced above would be set to {{math|''C'' {{=}} 1}}, and the integral would be restricted to the selected subregion of phase space.)

A simpler way to correct the overcounting is to integrate over all of phase space but to reduce the weight of each phase in order to exactly compensate the overcounting. This is accomplished by the factor {{math|''C''}} introduced above, which is a whole number that represents how many ways a physical state can be represented in phase space. Its value does not vary with the continuous canonical coordinates,<ref group=note>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 (mathematics)|measure]] zero) and so they do not contribute to any volume integral in phase space.</ref> so overcounting can be corrected simply by integrating over the full range of canonical coordinates, then dividing the result by the overcounting factor. However, {{math|''C''}} does vary strongly with discrete variables such as numbers of particles, and so it must be applied before summing over particle numbers.

As mentioned above, the classic example of this overcounting is for a fluid system containing various kinds of particles, where any two particles of the same kind are indistinguishable and exchangeable. When the state is written in terms of the particles' individual positions and momenta, then the overcounting related to the exchange of identical particles is corrected by using<ref name="gibbs"/>
<math display="block">C = N_1! N_2! \cdots N_s!.</math>
This is known as "correct Boltzmann counting".