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{{Use American English|date = February 2019}}
{{Short description|Function in thermodynamics and statistical physics}}
{{statistical mechanics}}
[[File:Gasfas.png | thumb | 220x124px | right | alt= The thermal motions of the atoms or molecules in a gas are allowed to move freely, and the interactions between the two (the gas and the atoms/molecules) can be neglected. | The thermal motions of the atoms or molecules in a gas are allowed to move freely, and the interactions between the two (the gas and the atoms/molecules) can be neglected.]]
In [[physics]], a '''partition function''' describes the [[statistics|statistical]] properties of a system in [[thermodynamic equilibrium]].{{Citation needed|reason=definition of partition function requires referencing|date=December 2016}} Partition functions are [[function (mathematics)|functions]] of the thermodynamic [[state function|state variables]], such as the [[temperature]] and [[volume]]. Most of the aggregate [[thermodynamics|thermodynamic]] variables of the system, such as the [[energy|total energy]], [[Thermodynamic free energy|free energy]], [[entropy]], and [[pressure]], can be expressed in terms of the partition function or its [[derivative]]s. The partition function is dimensionless.
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Initially, let us assume that a thermodynamically large system is in [[thermal contact]] with the environment, with a temperature ''T'', and both the volume of the system and the number of constituent particles are fixed. A collection of this kind of system comprises an ensemble called a [[canonical ensemble]]. The appropriate [[mathematical expression]] for the canonical partition function depends on the [[degrees of freedom]] of the system, whether the context is [[classical mechanics]] or [[quantum mechanics]], and whether the spectrum of states is [[Discrete mathematics|discrete]] or [[Probability distribution#Continuous probability distribution|continuous]].{{Citation needed|reason=definition of partition function requires referencing|date=December 2016}}
==== Classical discrete system ====
For a canonical ensemble that is classical and discrete, the canonical partition function is defined as
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* <math> i </math> is the index for the [[Microstate (statistical mechanics)|microstates]] of the system;
* <math> e </math> is [[e (mathematical constant)|Euler's number]];
* <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math> where <math>k_\text{B}</math> is the [[Boltzmann constant]];
* <math> E_i </math> is the total energy of the system in the respective [[Microstate (statistical mechanics)|microstate]].
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There are multiple approaches to deriving the partition function. The following derivation follows the more powerful and general [[information theory|information-theoretic]] [[Edwin Thompson Jaynes|Jaynesian]] [[maximum entropy thermodynamics|maximum entropy]] approach.
According to the [[second law of thermodynamics]], a system assumes a configuration of [[maximum entropy thermodynamics|maximum entropy]] at [[thermodynamic equilibrium]]
<math display="block"> S = - k_\text{B} \sum_i \rho_i \ln \rho_i </math>
subject to two physical constraints:
# The probabilities of all states add to unity ([[Probability axioms#Second axiom|second axiom of probability]]): <math display="block">▼
▲#The probabilities of all states add to unity ([[Probability axioms#Second axiom|second axiom of probability]]): <math display="block">
\sum_i \rho_i = 1.
</math>
# In the [[canonical ensemble]], the system is in [[thermal equilibrium]], so the average energy does not change over time; in other words, the average energy is
\langle E \rangle = \sum_i \rho_i E_i \equiv U
</math>
Applying [[calculus of variations|variational calculus]] with constraints (analogous in some sense to the method of [[Lagrange multipliers]]), we write the Lagrangian (or Lagrange function) <math> \mathcal{L} </math> as
<math display="block">
\mathcal{L} = \left( -k_\text{B} \sum_i \rho_i \ln \rho_i \right)
Varying and extremizing <math> \mathcal{L} </math> with respect to <math> \rho_i </math> leads to
<math display="block">\begin{align}
0 & \equiv \delta \mathcal{L} \\
&= \delta
&= \sum_i \
&= \sum_i \left[ \frac{\partial}{\partial \rho_i } \
&= \sum_i \
\end{align}</math>
Since this equation should hold for any variation <math> \delta ( \rho_i ) </math>, it implies that
<math display="block"> 0 \equiv - k_\text{B} \ln \rho_i - k_\text{B}
Isolating for <math> \rho_i </math> yields
<math display="block">\rho_i = \exp \left( \frac{-k_\text{B}
To obtain <math> \lambda_1 </math>, one substitutes the probability into the first constraint:
<math display="block">\begin{align}
1 &= \sum_i \rho_i \\
&= \exp \left( \frac{-k_\text{B}
\end{align}</math>
where '''<math> Z </math> is a
<math display="block">Z \equiv \sum_i \exp \left( - \frac{\lambda_2}{k_\text{B}} E_i \right) .</math>
Isolating for <math> \lambda_1 </math> yields <math> \lambda_1 =
Rewriting <math> \rho_i </math> in terms of <math> Z </math> gives
<math display="block"> \rho_i = \frac{1}{Z} \exp \left( - \frac{\lambda_2}{k_\text{B}} E_i \right) .</math>
Rewriting <math> S </math> in terms of <math> Z </math> gives
<math display="block">\begin{align}
S &= - k_\text{B} \sum_i \rho_i \ln \rho_i \\
&= - k_\text{B} \sum_i \rho_i \left( - \frac{\lambda_2}{k_\text{B}} E_i - \ln(Z) \right) \\
&=
&=
\end{align}</math>
To obtain <math> \lambda_2 </math>, we differentiate <math> S </math> with respect to the average energy <math> U </math> and apply the [[first law of thermodynamics]], <math> dU = T dS - P dV </math>:
<math display="block">\frac{dS}{dU} =
(Note that <math> \lambda_2 </math> and <math> Z </math> vary with <math> U </math> as well; however, using the chain rule and
<math display="block"> \frac{d}{d\lambda_2} \ln(Z) = - \frac{1}{k_\text{B}} \sum_i \rho_i E_i = - \frac{U}{k_\text{B}}, </math>
one can show that the additional contributions to this derivative cancel each other.)
Thus the canonical partition function <math> Z </math> becomes
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}}
==== Classical continuous system ====
In [[classical mechanics]], the [[Position (vector)|position]] and [[Momentum vector|momentum]] variables of a particle can vary continuously, so the set of microstates is actually [[uncountable set|uncountable]]. In ''classical'' statistical mechanics, it is rather inaccurate to express the partition function as a [[Sum (mathematics)|sum]] of discrete terms. In this case we must describe the partition function using an [[integral]] rather than a sum. For a canonical ensemble that is classical and continuous, the canonical partition function is defined as
<math display="block"> Z = \frac{1}{h^3} \int e^{-\beta H(q, p)} \,
where
* <math> h </math> is the [[Planck constant]];
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* <math> p </math> is the [[Canonical coordinates|canonical momentum]].
To make it into a dimensionless quantity, we must divide it by ''h'', which is some quantity with units of [[action (physics)|action]] (usually taken to be the [[Planck
For generalized cases, the partition function of <math> N </math> particles in <math> d </math>-dimensions is given by
====Classical continuous system (multiple identical particles)====▼
<math display="block"> Z = \frac{1}{
For a gas of <math> N </math> identical classical particles in three dimensions, the partition function is▼
▲<math display="block"> Z=\frac{1}{N!h^{3N}} \int \, \exp \left(-\beta \sum_{i=1}^N H(\textbf q_i, \textbf p_i) \right) \; \mathrm{d}^3 q_1 \cdots \mathrm{d}^3 q_N \, \mathrm{d}^3 p_1 \cdots \mathrm{d}^3 p_N </math>
▲==== Classical continuous system (multiple identical particles) ====
▲For a gas of <math> N </math> identical classical non-interacting particles in three dimensions, the partition function is
<math display="block"> Z=\frac{1}{N!h^{3N}} \int \, \exp \left(-\beta \sum_{i=1}^N H(\textbf q_i, \textbf p_i) \right) \; d^3 q_1 \cdots d^3 q_N \, d^3 p_1 \cdots d^3 p_N = \frac{Z_{\text{single}}^N}{N!}</math>
where
* <math> h </math> is the [[Planck constant]];
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* <math> q_i </math> is the [[Canonical coordinates|canonical position]] of the respective particle;
* <math> p_i </math> is the [[Canonical coordinates|canonical momentum]] of the respective particle;
* <math>
* <math> Z_{\text{single}} </math> is the classical continuous partition function of a single particle as given in the previous section.
The reason for the [[factorial]] factor ''N''! is discussed [[#Partition functions of subsystems|below]]. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not [[dimensionless]]. As stated in the previous section, to make it into a dimensionless quantity, we must divide it by ''h''<sup>3''N''</sup> (where ''h'' is usually taken to be the Planck
==== Quantum mechanical discrete system ====
For a canonical ensemble that is quantum mechanical and discrete, the canonical partition function is defined as the [[trace (linear algebra)|trace]] of the Boltzmann factor:
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The [[dimension]] of <math> e^{-\beta \hat{H}} </math> is the number of [[energy eigenstates]] of the system.
==== Quantum mechanical continuous system ====
For a canonical ensemble that is quantum mechanical and continuous, the canonical partition function is defined as
<math display="block"> Z = \frac{1}{h} \int \left\langle q, p
where:
* <math> h </math> is the [[Planck constant]];
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In systems with multiple [[quantum states]] ''s'' sharing the same energy ''E<sub>s</sub>'', it is said that the [[energy levels]] of the system are [[Degenerate energy levels|degenerate]]. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by ''j'') as follows:
<math display="block"> Z = \sum_j g_j \
where ''g<sub>j</sub>'' is the degeneracy factor, or number of quantum states ''s'' that have the same energy level defined by ''E<sub>j</sub>'' = ''E<sub>s</sub>''.
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where {{math|''Ĥ''}} is the [[Hamiltonian (quantum mechanics)|quantum Hamiltonian operator]]. The exponential of an operator can be defined using the [[Characterizations of the exponential function|exponential power series]].
The classical form of ''Z'' is recovered when the trace is expressed in terms of [[coherent state]]s<ref>{{cite book |first1=John R. |last1=Klauder |first2=Bo-Sture |last2=Skagerstam |title=Coherent States: Applications in Physics and Mathematical Physics |publisher=World Scientific |date=1985 |pages=71–73 |isbn=978-9971-966-52-2 }}</ref> and when quantum-mechanical [[uncertainty principle|uncertainties]] in the position and momentum of a particle are regarded as negligible. Formally, using [[bra–ket notation]], one inserts under the trace for each degree of freedom the identity:
<math display="block"> \boldsymbol{1} = \int |x, p\rangle \langle x,p| \frac{dx \,dp}{h},</math>
where {{ket|''x'', ''p''}} is a [[Normalizing constant|normalised]] [[Wave packet#Gaussian wave packets in quantum mechanics|Gaussian wavepacket]] centered at position ''x'' and momentum ''p''. Thus
<math display="block">
Z = \int \operatorname{tr} \left( e^{-\beta\hat{H}} |x, p\rangle \langle x, p| \right) \frac{dx \,dp}{h}
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For simplicity, we will use the discrete form of the partition function in this section. Our results will apply equally well to the continuous form.
Consider a system ''S'' embedded into a [[heat bath]] ''B''. Let the total [[energy]] of both systems be ''E''. Let ''p<sub>i</sub>'' denote the [[probability]] that the system ''S'' is in a particular [[Microstate (statistical mechanics)|microstate]], ''i'', with energy ''E<sub>i</sub>''. According to the [[Statistical mechanics#Fundamental postulate|fundamental postulate of statistical mechanics]] (which states that all attainable microstates of a system are equally probable), the probability ''p<sub>i</sub>'' will be
<math display="block">p_i = \frac{\Omega_B(E - E_i)}{\Omega_{(S,B)}(E)}.</math>
Assuming that the heat bath's internal energy is much larger than the energy of ''S'' ({{nowrap|''E'' ≫ ''E<sub>i</sub>''}}), we can [[Taylor expansion|Taylor-expand]] <math>\Omega_B</math> to first order in ''E<sub>i</sub>'' and use the thermodynamic relation <math>\partial S_B/\partial E = 1/T</math>, where here <math>S_B</math>, <math>T</math> are the entropy and temperature of the bath respectively:
<math display="block">\begin{align}
k \ln p_i &= k \ln \Omega_B(E - E_i) - k \ln \Omega_{(S,B)}(E) \\[5pt]
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In order to demonstrate the usefulness of the partition function, let us calculate the thermodynamic value of the total energy. This is simply the [[expected value]], or [[ensemble average]] for the energy, which is the sum of the microstate energies weighted by their probabilities:
<math display="block">\begin{align}
\langle E \rangle = \sum_s E_s P_s &= \frac{1}{Z} \sum_s E_s e^{- \beta E_s} \\[1ex] \end{align}
</math>
or, equivalently,
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The [[variance]] in the energy (or "energy fluctuation") is
<math display="block">\left\langle (\Delta E)^2 \right\rangle \equiv \left\langle (E - \langle E\rangle)^2 \right\rangle
= \frac{\partial^2 \ln Z}{\partial \beta^2}.</math>
The [[heat capacity]] is
<math display="block">C_v = \frac{\partial \langle E \rangle}{\partial T} = \frac{1}{k_\text{B} T^2} \left\langle (\Delta E)^2 \right\rangle.</math>
In general, consider the [[extensive variable]] ''X'' and [[intensive variable]] ''Y'' where ''X'' and ''Y'' form a pair of [[conjugate variables]]. In ensembles where ''Y'' is fixed (and ''X'' is allowed to fluctuate), then the average value of ''X'' will be:
<math display="block">\langle X \rangle = \pm \frac{\partial \ln Z}{\partial \beta Y}.</math>
The sign will depend on the specific definitions of the variables ''X'' and ''Y''. An example would be ''X'' = volume and ''Y'' = pressure. Additionally, the variance in ''X'' will be
<math display="block">\left\langle (\Delta X)^2 \right\rangle \equiv \left\langle (X - \langle
X\rangle)^2 \right\rangle = \frac{\partial \langle X \rangle}{\partial \beta Y} = \frac{\partial^2 \ln Z}{\partial (\beta Y)^2}.</math>
In the special case of [[entropy]], entropy is given by
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Furthermore, the heat capacity can be expressed as
<math display="block">
=== Partition functions of subsystems ===
Suppose a system is subdivided into ''N'' sub-systems with negligible interaction energy, that is, we can assume the particles are essentially non-interacting. If the partition functions of the sub-systems are ''ζ''<sub>1</sub>, ''ζ''<sub>2</sub>, ..., ''ζ''<sub>N</sub>, then the partition function of the entire system is the ''product'' of the individual partition functions:
<math display="block">Z = \prod_{j=1}^{N} \zeta_j.</math>
If the sub-systems have the same physical properties, then their partition functions are equal, ''ζ''<sub>1</sub> = ''ζ''<sub>2</sub> = ... = ''ζ'', in which case <math display="block">Z = \zeta^N.</math>
However, there is a well-known exception to this rule. If the sub-systems are actually [[identical particles]], in the [[quantum mechanics|quantum mechanical]] sense that they are impossible to distinguish even in principle, the total partition function must be divided by a ''N''! (''N''
<math display="block">Z = \frac{\zeta^N}{N!}.</math>
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= 1. </math>
This is the reason for calling ''Z'' the "partition function": it encodes how the probabilities are partitioned among the different microstates, based on their individual energies. Other partition functions for different ensembles divide up the probabilities based on other macrostate variables. As an example: the partition function for the [[isothermal-isobaric ensemble]], the [[Boltzmann distribution#Generalized Boltzmann distribution|generalized Boltzmann distribution]], divides up probabilities based on particle number, pressure, and temperature. The energy is replaced by the characteristic potential of that ensemble, the [[Gibbs Free Energy]]. The letter ''Z'' stands for the [[German language|German]] word ''Zustandssumme'', "sum over states". The usefulness of the partition function stems from the fact that
== Grand canonical partition function ==
{{Main|Grand canonical ensemble}}
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The grand canonical partition function, denoted by <math>\mathcal{Z}</math>, is the following sum over [[microstate (statistical mechanics)|microstates]]
Here, each microstate is labelled by <math>i</math>, and has total particle number <math>N_i</math> and total energy <math>E_i</math>. This partition function is closely related to the [[grand potential]], <math>\Phi_{\rm G}</math>, by the relation
This can be contrasted to the canonical partition function above, which is related instead to the [[Helmholtz free energy]].
It is important to note that the number of microstates in the grand canonical ensemble may be much larger than in the canonical ensemble, since here we consider not only variations in energy but also in particle number. Again, the utility of the grand canonical partition function is that it is related to the probability that the system is in state <math>i</math>:
An important application of the grand canonical ensemble is in deriving exactly the statistics of a non-interacting many-body quantum gas ([[Fermi–Dirac statistics]] for fermions, [[Bose–Einstein statistics]] for bosons), however it is much more generally applicable than that. The grand canonical ensemble may also be used to describe classical systems, or even interacting quantum gases.
The grand partition function is sometimes written (equivalently) in terms of alternate variables as<ref>{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc. }}</ref>
where <math>z \equiv \exp(\mu/
== See also ==
* [[Partition function (mathematics)]]
* [[Partition function (quantum field theory)]]
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* [[Widom insertion method]]
== References ==
{{reflist}}
{{refbegin}}
* {{cite book |last=Huang
* {{cite book |first=A. |last=Isihara
* {{cite web |last=Kelly
* {{cite book |first=L. D. |last=Landau
* {{cite web |last=Vu-Quoc
{{refend}}
{{Statistical mechanics topics}}
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