Editing Microcanonical ensemble (section)

== Precise expressions for the ensemble ==

The precise mathematical expression for a statistical ensemble depends on the kind of mechanics under consideration – quantum or classical – since the notion of a "microstate" is considerably different in these two cases. In quantum mechanics, [[Matrix diagonalization|diagonalization]] provides a discrete set of [[microstate (statistical mechanics)|microstates]] with specific energies. The classical mechanical case involves instead an integral over canonical [[phase space]], and the size of microstates in phase space can be chosen somewhat arbitrarily.

To construct the microcanonical ensemble, it is necessary in both types of mechanics to first specify a range of energy. In the expressions below the function <math>f{\left(\tfrac{H - E}{\omega}\right)}</math> (a function of {{math|''H''}}, peaking at {{math|''E''}} with width {{math|''ω''}}) will be used to represent the range of energy in which to include states. An example of this function would be<ref name="gibbs"/>
<math display="block">f(x) = \begin{cases} 1, & \text{if}~|x| < \tfrac{1}{2}, \\  0, & \text{otherwise.} \end{cases}</math>
or, more smoothly,
<math display="block">f(x) = e^{-\pi x^2}.</math>

=== Quantum mechanical ===

{{multiple image
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| header    = Example of microcanonical ensemble for a quantum system consisting of one particle in a potential well.
| footer    = {{small|The particle's Hamiltonian is [[Schrödinger equation|Schrödinger]]-type, {{math|1=''Ĥ'' = ''U''(''x'') + ''p''<sup>2</sup>/2''m''}} (the potential {{math|''U''(''x'')}} is plotted as a red curve). Each panel shows an energy-position plot with the various stationary states, along with a side plot showing the distribution of states in energy.}}
<!-- Image 1 -->
| image1    = Ensemble quantum 1DOF all states.png
| width1    =
| alt1      =
| caption1  = Plot of all possible states of this system. The available stationary states displayed as horizontal bars of varying darkness according to {{math|{{abs|''ψ''<sub>''i''</sub>(x)}}<sup>2</sup>}}.
<!-- Image 2 -->
| image2    = Ensemble quantum 1DOF microcanonical.png
| width2    =
| alt2      =
| caption2  = An ensemble containing only those states within a narrow interval of energy. As the energy width is taken to zero, a microcanonical ensemble is obtained (provided the interval contains at least one state).
}}

{{details|topic=the representation of ensembles in quantum mechanics|Statistical ensemble (mathematical physics)}}

A statistical ensemble in quantum mechanics is represented by a [[density matrix]], denoted by <math>\hat\rho</math>. The microcanonical ensemble can be written using [[bra–ket notation]], in terms of the system's [[stationary state|energy eigenstates]] and energy eigenvalues. Given a complete basis of energy eigenstates {{math|{{ket|''ψ''<sub>''i''</sub>}}}}, indexed by {{math|''i''}}, the microcanonical ensemble is{{citation needed|date=November 2013}}
<math display="block">\hat\rho = \frac{1}{W} \sum_i f{\left(\tfrac{H_i - E}{\omega}\right)} \left|\psi_i\right\rangle \left\langle \psi_i \right|,</math>
where the {{math|''H''<sub>''i''</sub>}} are the energy eigenvalues determined by <math>\hat H |\psi_i\rangle = H_i |\psi_i\rangle</math> (here {{math|''Ĥ''}} is the system's total energy operator, i. e., [[Hamiltonian (quantum mechanics)|Hamiltonian operator]]). The value of {{math|''W''}} is determined by demanding that <math>\hat\rho</math> is a normalized density matrix, and so
<math display="block">W = \sum_i f{\left(\tfrac{H_i - E}{\omega}\right)}.</math>
The state volume function (used to calculate entropy) is given by
<math display="block">v(E) = \sum_{H_i < E} 1.</math>

The microcanonical ensemble is defined by taking the limit of the density matrix as the energy width goes to zero, however a problematic situation occurs once the energy width becomes smaller than the spacing between energy levels. For very small energy width, the ensemble does not exist at all for most values of {{math|''E''}}, since no states fall within the range. When the ensemble does exist, it typically only contains one ([[Kramers theorem|or two]]) states, since in a complex system the energy levels are only ever equal by accident (see [[random matrix theory]] for more discussion on this point). Moreover, the state-volume function also increases only in discrete increments, and so its derivative is only ever infinite or zero, making it difficult to define the density of states. This problem can be solved by not taking the energy range completely to zero and smoothing the state-volume function, however this makes the definition of the ensemble more complicated, since it becomes then necessary to specify the energy range in addition to other variables (together, an {{math|''NVEω''}} ensemble).

=== Classical mechanical ===

{{multiple image
<!-- Essential parameters -->
| align     = right
| direction = horizontal
| width     = 220
| header    = Example of microcanonical ensemble for a classical system consisting of one particle in a potential well.
| footer    = Each panel shows [[phase space]] (upper graph) and energy-position space (lower graph). The particle's Hamiltonian is {{math|1=''H'' = ''U''(''x'') + ''p''<sup>2</sup>/2''m''}}, with the potential {{math|''U''(''x'')}} shown as a red curve. The side plot shows the distribution of states in energy.
<!-- Image 1 -->
| image1    = Ensemble classical 1DOF all states.png
| width1    =
| alt1      =
| caption1  = Plot of all possible states of this system. The available physical states are evenly distributed in phase space, but with an uneven distribution in energy; the side-plot displays {{math|''dv''/''dE''}}.
<!-- Image 2 -->
| image2    = Ensemble classical 1DOF microcanonical.png
| width2    =
| alt2      =
| caption2  = An ensemble restricted to only those states within a narrow interval of energy. This ensemble appears as a thin shell in phase space. As the energy width is taken to zero, a microcanonical ensemble is obtained.
}}

{{details|topic=the representation of ensembles in classical mechanics|Statistical ensemble (mathematical physics)}}

In classical mechanics, an ensemble is 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>)}} defined over the system's [[phase space]].<ref name="gibbs"/> 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 probability density function for the microcanonical ensemble is:
<math display="block">\rho = \frac{1}{h^n C} \frac{1}{W} f{\left(\tfrac{H-E}{\omega}\right)},</math>
where
* {{math|''H''}} is the total energy ([[Hamiltonian mechanics|Hamiltonian]]) of the system, a function of the phase {{math|(''p''<sub>1</sub>, … ''q''<sub>''n''</sub>)}},
* {{math|''h''}} is an arbitrary but predetermined constant with the units of {{math|energy×time}}, setting the extent of one microstate and providing correct dimensions to {{math|''ρ''}}.<ref group=note>(Historical note) Gibbs' original ensemble effectively set {{math|1=''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, often used for particle systems where identical particles are able to change place with each other.<ref group=note>In a system of {{math|''N''}} identical particles, {{math|1=''C'' = ''N''!}} ([[factorial]] of {{math|''N''}}). This factor corrects the overcounting in phase space due to identical physical states being found in multiple locations. See the [[Statistical ensemble (mathematical physics)#Correcting overcounting in phase space|statistical ensemble]] article for more information on this overcounting.</ref>

Again, the value of {{math|''W''}} is determined by demanding that {{math|''ρ''}} is a normalized probability density function:
<math display="block">W = \int \cdots \int \frac{1}{h^n C} f{\left(\tfrac{H-E}{\omega}\right)} \, dp_1 \cdots dq_n </math>
This integral is taken over the entire [[phase space]]. The state volume function (used to calculate entropy) is defined by
<math display="block">v(E) = \int \cdots \int_{H < E} \frac{1}{h^n C} \, dp_1 \cdots dq_n .</math>

As the energy width {{math|''ω''}} is taken to zero, the value of {{math|''W''}} decreases in proportion to {{math|''ω''}} as {{math|1=''W'' = ''ω'' (''dv''/''dE'')}}.

Based on the above definition, the microcanonical ensemble can be visualized as an infinitesimally thin shell in phase space, centered on a constant-energy surface. Although the microcanonical ensemble is confined to this surface, it is not necessarily uniformly distributed over that surface: if the gradient of energy in phase space varies, then the microcanonical ensemble is "thicker" (more concentrated) in some parts of the surface than others. This feature is an unavoidable consequence of requiring that the microcanonical ensemble is a steady-state ensemble.