Editing Ensemble (mathematical physics) (section)

== Representations ==

The precise mathematical expression for a statistical ensemble has a distinct form depending on the type of mechanics under consideration (quantum or classical). In the classical case, the ensemble is a [[probability distribution]] over the microstates.  In quantum mechanics, this notion, due to [[John von Neumann|von Neumann]], is a way of assigning a probability distribution over the results of each [[complete set of commuting observables]]. 
In classical mechanics, the ensemble is instead written as a probability distribution in [[phase space]]; the microstates are the result of partitioning phase space into equal-sized units, although the size of these units can be chosen somewhat arbitrarily.

=== Requirements for representations ===

Putting aside for the moment the question of how statistical ensembles are generated [[operational definition|operationally]], we should be able to perform the following two operations on ensembles ''A'', ''B'' of the same system:
* Test whether ''A'', ''B'' are statistically equivalent.
* If ''p'' is a real number such that {{math|0 < ''p'' < 1}}, then produce a new ensemble by probabilistic sampling from ''A'' with probability ''p'' and from ''B'' with probability {{math|1 − ''p''}}.

Under certain conditions, therefore, [[equivalence class]]es of statistical ensembles have the structure of a convex set.

=== Quantum mechanical ===
{{main|Density matrix}}
A statistical ensemble in quantum mechanics (also known as a mixed state) is most often represented by a [[density matrix]], denoted by <math>\hat\rho</math>. The density matrix provides a fully general tool that can incorporate both quantum uncertainties (present even if the state of the system were completely known) and classical uncertainties (due to a lack of knowledge) in a unified manner. Any physical observable {{mvar|X}} in quantum mechanics can be written as an operator, <math>\hat X</math>. The expectation value of this operator on the statistical ensemble <math>\rho</math> is given by the following [[trace (linear algebra)|trace]]:
<math display="block">\langle X \rangle = \operatorname{Tr}(\hat X \rho).</math>
This can be used to evaluate averages (operator <math>\hat X</math>), [[variance]]s (using operator <math>\hat X^2</math>), [[covariance]]s (using operator <math>\hat X \hat Y</math>), etc. The density matrix must always have a trace of 1: <math>\operatorname{Tr}{\hat\rho} = 1</math> (this essentially is the condition that the probabilities must add up to one).

In general, the ensemble evolves over time according to the [[von Neumann equation]].

Equilibrium ensembles (those that do not evolve over time, <math>d\hat\rho / dt = 0</math>) can be written solely as a function of conserved variables. For example, the [[microcanonical ensemble]] and [[canonical ensemble]] are strictly functions of the total energy, which is measured by the total energy operator <math>\hat H</math> (Hamiltonian). The grand canonical ensemble is additionally a function of the particle number, measured by the total particle number operator <math>\hat N</math>. Such equilibrium ensembles are a [[diagonal matrix]] in the orthogonal basis of states that simultaneously diagonalize each conserved variable. In [[bra–ket notation]], the density matrix is
<math display="block">\hat\rho = \sum_i P_i |\psi_i\rangle \langle\psi_i|,</math>
where the {{math|{{ket|''ψ''<sub>''i''</sub>}}}}, indexed by {{mvar|i}}, are the elements of a complete and orthogonal basis. (Note that in other bases, the density matrix is not necessarily diagonal.)

=== Classical mechanical ===

[[File:Hamiltonian flow classical.gif|frame|Evolution of an ensemble of [[Hamiltonian mechanics|classical]] systems in [[phase space]] (top). Each system consists of one massive particle in a one-dimensional [[potential well]] (red curve, lower figure). The initially compact ensemble becomes swirled up over time.]]

In classical mechanics, an ensemble is represented by a probability density function defined over the system's [[phase space]].<ref name="gibbs"/> While an individual system evolves according to [[Hamilton's equations]], the density function (the ensemble) evolves over time according to [[Liouville's theorem (Hamiltonian)|Liouville's equation]].

In a [[Hamiltonian mechanics|mechanical system]] with a defined number of parts, the phase space has {{math|''n''}} [[generalized coordinates]] called {{math|''q''<sub>1</sub>, ... ''q''<sub>''n''</sub>}}, and {{math|''n''}} associated [[canonical momentum|canonical momenta]] called {{math|''p''<sub>1</sub>, ... ''p''<sub>''n''</sub>}}. The ensemble is then represented by a [[joint probability density function]] {{math|''ρ''(''p''<sub>1</sub>, ... ''p''<sub>''n''</sub>, ''q''<sub>1</sub>, ... ''q''<sub>''n''</sub>)}}.

If the number of parts in the system is allowed to vary among the systems in the ensemble (as in a grand ensemble where the number of particles is a random quantity), then it is a probability distribution over an extended phase space that includes further variables such as particle numbers {{math|''N''<sub>1</sub>}} (first kind of particle), {{math|''N''<sub>2</sub>}} (second kind of particle), and so on up to {{math|''N''<sub>''s''</sub>}} (the last kind of particle; {{math|''s''}} is how many different kinds of particles there are). The ensemble is then represented by a [[joint probability density function]] {{math|''ρ''(''N''<sub>1</sub>, ... ''N''<sub>''s''</sub>, ''p''<sub>1</sub>, ... ''p''<sub>''n''</sub>, ''q''<sub>1</sub>, ... ''q''<sub>''n''</sub>)}}. The number of coordinates {{math|''n''}} varies with the numbers of particles.

Any mechanical quantity {{math|''X''}} can be written as a function of the system's phase. The expectation value of any such quantity is given by an integral over the entire phase space of this quantity weighted by {{math|''ρ''}}:
<math display="block">\langle X \rangle = \sum_{N_1 = 0}^{\infty} \cdots \sum_{N_s = 0}^{\infty} \int \cdots \int \rho X \, dp_1 \cdots dq_n.</math>
The condition of probability normalization applies, requiring
<math display="block">\sum_{N_1 = 0}^{\infty} \cdots \sum_{N_s = 0}^{\infty} \int \cdots \int \rho \, dp_1 \cdots dq_n = 1.</math>

Phase space is a continuous space containing an infinite number of distinct physical states within any small region. In order to connect the probability ''density'' in phase space to a probability ''distribution'' over microstates, it is necessary to somehow partition the phase space into blocks that are distributed representing the different states of the system in a fair way. It turns out that the correct way to do this simply results in equal-sized blocks of canonical phase space, and so a microstate in classical mechanics is an extended region in the phase space of canonical coordinates that has a particular volume.<ref group=note>This equal-volume partitioning is a consequence of [[Liouville's theorem (Hamiltonian)|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.</ref> In particular, the probability density function in phase space, {{math|''ρ''}}, is related to the probability distribution over microstates, {{math|''P''}} by a factor
<math display="block">\rho = \frac{1}{h^n C} P,</math>
where
* {{math|''h''}} is an arbitrary but predetermined constant with the units of {{math|energy×time}}, setting the extent of the microstate and providing correct dimensions to {{math|''ρ''}}.<ref group=note>(Historical note) Gibbs' original ensemble effectively set {{math|''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, {{math|''h''}} is often taken to be equal to the [[Planck constant]] in order to obtain a semiclassical correspondence with quantum mechanics.</ref>
* {{math|''C''}} is an overcounting correction factor (see below), generally dependent on the number of particles and similar concerns.
Since {{math|''h''}} can be chosen arbitrarily, the notional size of a microstate is also arbitrary. Still, the value of {{math|''h''}} influences the offsets of quantities such as entropy and chemical potential, and so it is important to be consistent with the value of {{math|''h''}} when comparing different systems.

==== 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".