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Canonical ensemble
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=== Classical mechanical === {{multiple image <!-- Essential parameters --> | align = right | direction = horizontal | width = 220 | header = Example of canonical 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|''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 canonical.png | width2 = | alt2 = | caption2 = A canonical ensemble for this system, for the temperature shown. The states are weighted exponentially in energy. }} {{details|topic=the representation of ensembles in classical mechanics|Statistical ensemble (mathematical physics)}} In classical mechanics, a statistical ensemble is instead represented by a [[joint probability density function]] in the system's [[phase space]], {{math|''ρ''(''p''<sub>1</sub>, … ''p''<sub>''n''</sub>, ''q''<sub>1</sub>, … ''q''<sub>''n''</sub>)}}, where the {{math|''p''<sub>1</sub>, … ''p''<sub>''n''</sub>}} and {{math|''q''<sub>1</sub>, … ''q''<sub>''n''</sub>}} are the [[canonical coordinates]] (generalized momenta and generalized coordinates) of the system's internal degrees of freedom. In a system of particles, the number of degrees of freedom {{math|''n''}} depends on the number of particles {{math|''N''}} in a way that depends on the physical situation. For a three-dimensional monoatomic gas (not molecules), {{math|''n'' {{=}} 3''N''}}. In diatomic gases there will also be rotational and vibrational degrees of freedom. The probability density function for the canonical ensemble is: : <math>\rho = \frac{1}{h^n C} e^{\frac{F - E}{k T}},</math> where * {{math|''E''}} is the energy 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|''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|''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> * {{math|''F''}} provides a normalizing factor and is also the characteristic state function, the free energy. Again, the value of {{math|''F''}} is determined by demanding that {{math|''ρ''}} is a normalized probability density function: : <math>e^{-\frac{F}{k T}} = \int \ldots \int \frac{1}{h^n C} e^{\frac{- E}{k T}} \, dp_1 \ldots dq_n </math> This integral is taken over the entire [[phase space]]. In other words, a microstate in classical mechanics is a phase space region, and this region has volume {{math|''h<sup>n</sup>C''}}. This means that each microstate spans a range of energy, however this range can be made arbitrarily narrow by choosing {{math|''h''}} to be very small. The phase space integral can be converted into a summation over microstates, once phase space has been finely divided to a sufficient degree.
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