Editing Microcanonical ensemble (section)

=== Classical mechanical ===

{{multiple image
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| 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.
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| image1    = Ensemble classical 1DOF all states.png
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| 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''}}.
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| image2    = Ensemble classical 1DOF microcanonical.png
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| 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.
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{{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.