Editing Grand canonical ensemble (section)

== Example ensembles ==

The usefulness of the grand canonical ensemble is illustrated in the examples below. In each case the grand potential is calculated on the basis of the relationship
<math display="block">\Omega = -kT \ln \left(\sum_\text{microstates} e^{{(\mu N - E)}/{(kT)}} \right)</math>
which is required for the microstates' probabilities to add up to 1.

=== Statistics of noninteracting particles ===

==== Bosons and fermions (quantum) ====

In the special case of a quantum system of many ''non-interacting'' particles, the thermodynamics are simple to compute.<ref>{{cite book |title=Statistical Mechanics |last1=Srivastava |first1=R. K. |last2=Ashok |first2=J. |year=2005 |publisher=PHI Learning Pvt. Ltd. |isbn=9788120327825 |location=[[New Delhi]] }}</ref>
Since the particles are non-interacting, one can compute a series of single-particle [[stationary state]]s, each of which represent a separable part that can be included into the total quantum state of the system.
For now let us refer to these single-particle stationary states as ''orbitals'' (to avoid confusing these "states" with the total many-body state), with the provision that each possible internal particle property ([[electron spin|spin]] or [[polarization (waves)|polarization]]) counts as a separate orbital.
Each orbital may be occupied by a particle (or particles), or may be empty.

Since the particles are non-interacting, we may take the viewpoint that ''each orbital forms a separate thermodynamic system''.
Thus each orbital is a grand canonical ensemble unto itself, one so simple that its statistics can be immediately derived here. Focusing on just one orbital labelled {{math|''i''}}, the total energy for a [[microstate (statistical mechanics)|microstate]] of {{math|''N''<sub>''i''</sub>}} particles in this orbital will be {{math|''N''<sub>''i''</sub>''ϵ''<sub>''i''</sub>}}, where {{math|''ϵ''<sub>''i''</sub>}} is the characteristic energy level of that orbital. The grand potential for the orbital is given by one of two forms, depending on whether the orbital is bosonic or fermionic:
{{unordered list
|1= For [[fermion]]s, the [[Pauli exclusion principle]] allows only two microstates for the orbital (occupation of 0 or 1), giving a two-term series
<math display="block">\begin{align}
\Omega_i & = -kT \ln \Big( \sum_{N_i=0}^{1} e^{{(N_i\mu - N_i\epsilon_i)}/{(k T)}}\Big) \\
 & = - kT \ln\Big(1 + e^{{(\mu - \epsilon_i)}/{(k T)}}\Big)
\end{align}</math>
|2= For [[boson]]s, {{math|''N''<sub>''i''</sub>}} may be any nonnegative integer and each value of {{math|''N''<sub>''i''</sub>}} counts as one microstate due to the [[Identical particles|indistinguishability of particles]], leading to a [[geometric series]]:
<math display="block">\begin{align}
\Omega_i & = -kT \ln \Big( \sum_{N_i=0}^{\infty} e^{{(N_i\mu - N_i\epsilon_i)}/{(k T)}} \Big) \\
 & = + kT\ln\Big(1 - e^{{(\mu - \epsilon_i)}/{(k T)}}\Big).
\end{align}</math>
}}
In each case the value <math>\textstyle\langle N_i\rangle = -\tfrac{\partial \Omega_i}{\partial \mu}</math> gives the thermodynamic average number of particles on the orbital: the [[Fermi–Dirac distribution]] for fermions, and the [[Bose–Einstein distribution]] for bosons.
Considering again the entire system, the total grand potential is found by adding up the {{math|&Omega;<sub>''i''</sub>}} for all orbitals.

==== Indistinguishable classical particles ====
In classical mechanics it is also possible to consider indistinguishable particles (in fact, indistinguishability is a prerequisite for defining a chemical potential in a consistent manner; all particles of a given kind must be interchangeable<ref name="gibbs"/>). We can consider a region of the single-particle phase space with approximately uniform energy {{math|''ϵ''<sub>''i''</sub>}} to be an "orbital" labelled {{math|''i''}}.

Two complications arise since this orbital actually encompasses many (infinite) distinct states. Briefly:

* An overcounting correction of {{math|1/''N''<sub>''i''</sub>!}} is needed since the ''many''-particle phase space contains {{math|''N''<sub>''i''</sub>!}} copies of the same actual state (formed by the permutation of the particles' different exact states).
* The chosen width of the orbital is arbitrary, thus there is a further proportionality factor that is independent of {{math|''N''<sub>''i''</sub>}} .

Due to the {{math|1/''N''<sub>''i''</sub>!}} overcounting correction, the summation now takes the form of an exponential [[power series]],
<math display="block">\begin{align}
\Omega_i & \propto -kT \ln \left( \sum_{N_i=0}^{\infty} \frac{1}{N_i!} e^{{(N_i\mu - N_i\epsilon_i)}/{(k T)}}\right) \\[1ex]
 & \propto -kT \ln \left( e^{e^{{(\mu - \epsilon_i)}/{(k T)}}}\right) \\[1ex]
 & \propto - kT e^{\frac{\mu - \epsilon_i}{k T}},
\end{align}</math>
the value <math>\scriptstyle\langle N_i\rangle \propto -\tfrac{\partial \Omega_i}{\partial \mu}</math> corresponding to [[Maxwell–Boltzmann statistics]].

=== Ionization of an isolated atom ===

The grand canonical ensemble can be used to predict whether an atom prefers to be in a neutral state or ionized state.
An atom is able to exist in ionized states with more or fewer electrons compared to neutral. As shown below, ionized states may be thermodynamically preferred depending on the environment.
Consider a simplified model where the atom can be in a neutral state or in one of two ionized states (a detailed calculation also includes excited states and the degeneracy factors of the states<ref name="saha">{{cite journal|doi=10.1119/1.1934002
|title=On the Use of Grand Ensembles in Statistical Mechanics: A New Derivation of Saha's Formula
|date=1955
|last1=Ter Haar
|first1=D.
|journal=American Journal of Physics
|volume=23
|issue=6
|pages=326–331
}}</ref><ref name="dopants">{{cite book|title=Semiconductor Physics and Applications|last1=Balkanski |first1=M.|last2=Wallis | first2=R.F.|year=2000|publisher=Oxford University Press|isbn=0198517408}}</ref>):
* charge neutral state, with {{math|''N''<sub>0</sub>}} electrons and energy {{math|''E''<sub>0</sub>}}.
* an [[oxidized]] state ({{math|''N''<sub>0</sub> − 1}} electrons) with energy {{math|''E''<sub>0</sub> + Δ''E''<sub>I</sub> + ''qϕ''}}
* a [[redox|reduced]] state ({{math|''N''<sub>0</sub> + 1}} electrons) with energy {{math|''E''<sub>0</sub> − Δ''E''<sub>A</sub> − ''qϕ''}}
Here {{math|Δ''E''<sub>I</sub>}} and {{math|Δ''E''<sub>A</sub>}} are the atom's [[ionization energy]] and [[electron affinity]], respectively; {{math|''ϕ''}} is the local [[electrostatic potential]] in the vacuum nearby the atom, and {{math|−''q''}} is the [[electron charge]].

The grand potential in this case is thus determined by
<math display="block"> \begin{align}
\Omega
& = -kT \ln \Big(e^{{(\mu N_0 - E_0)}/{(k T)}} + e^{{(\mu N_0 - \mu - E_0 - \Delta E_{\rm I} - q\phi)}/{(k T)}} + e^{{(\mu N_0 + \mu - E_0 + \Delta E_{\rm A} + q\phi)}/{(k T)}}\Big). \\
& = E_0 - \mu N_0 -kT \ln \Big( 1 + e^{{(-\mu - \Delta E_{\rm I} - q\phi)}/{(k T)}} + e^{{(\mu + \Delta E_{\rm A} + q\phi)}/{(k T)}}\Big). \\
\end{align} </math>
<!-- & = -kT \ln \Big(e^{{(\mu N_0 - E_0)}/{(k T)}} \Big[ 1 + e^{{(-\mu - \Delta E_{\rm I} - q\phi)}/{(k T)}} + e^{{(\mu + \Delta E_{\rm A} + q\phi)}/{(k T)}}\Big]\Big). \\ -->
The quantity {{math|−''qϕ'' − ''µ''}} is critical in this case, for determining the balance between the various states. This value is determined by the environment around the atom.

[[File:Surface ionization of cesium.svg|thumb|Surface ionization effect in a vaporized [[cesium]] atom at 1500 K, calculated using the method in this section (also including [[Degeneracy (quantum mechanics)|degeneracy]]). Y-axis: average number of electrons; the atom is neutral when it has 55 electrons. X-axis: energy variable, which is equal to the surface [[work function]].]]

* If a lone atom is placed in metal box, then {{math|−''qϕ'' − ''µ'' {{=}} ''W''}}, the [[work function]] of the box lining material. If <math>W > \Delta E_{\rm I}</math> then the atom prefers to exist as a positive ion. This spontaneous [[surface ionization]] effect has been used as a cesium [[ion source]].<ref>{{Cite journal | last1 = Alton | first1 = G. D. | title = Characterization of a cesium surface ionization source with a porous tungsten ionizer. I | doi = 10.1063/1.1139776 | journal = Review of Scientific Instruments | volume = 59 | issue = 7 | pages = 1039–1044 | year = 1988 |bibcode = 1988RScI...59.1039A | url = https://zenodo.org/record/1231832 }}</ref>
* If the atom is surrounded by an ideal electron gas with density <math>n_{\rm e}</math>, then <math> \exp((\mu + q\phi)/kT) = n_{\rm e}\lambda_{\rm th}^3 / 2</math>, and substituting this in yields the [[Saha ionization equation]].<ref name="saha"/>
* In [[semiconductor]]s, where the ionization of a [[dopant]] atom is well described by this ensemble.<ref name="dopants"/> In the semiconductor, the [[conduction band]] edge {{math|''ϵ''<sub>C</sub>}} plays the role of the vacuum energy level (replacing {{math|−''qϕ''}}), and {{math|''µ''}} is known as the [[Fermi level]]. Of course, the ionization energy and electron affinity of the dopant atom are strongly modified relative to their vacuum values. A typical donor dopant in silicon, phosphorus, has {{nowrap|1={{math|Δ''E''<sub>I</sub>}} = {{val|45|u=meV}}}}.<ref>{{Cite web|url=http://www.iue.tuwien.ac.at/phd/wittmann/node7.html|title=2. Semiconductor Doping Technology}}</ref>