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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 Each partition function is constructed to represent a particular [[statistical ensemble]] (which, in turn, corresponds to a particular [[Thermodynamic free energy|free energy]]). The most common statistical ensembles have named partition functions. The '''canonical partition function''' applies to a [[canonical ensemble]], in which the system is allowed to exchange [[heat]] with the [[Environment (systems)|environment]] at fixed temperature, volume, and [[number of particles]]. The '''grand canonical partition function''' applies to a [[grand canonical ensemble]], in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and [[chemical potential]]. Other types of partition functions can be defined for different circumstances; see [[partition function (mathematics)]] for generalizations. The partition function has many physical meanings, as discussed in [[#Meaning and significance|Meaning and significance]].
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=== Definition ===
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
For a canonical ensemble that is classical and discrete, the canonical partition function is defined as
<math display="block"> Z = \sum_i e^{-\beta E_i}, </math>
where
* <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]].
The [[Exponential function|exponential]] factor <math> e^{-\beta E_i} </math> is otherwise known as the [[Boltzmann factor]].
{{math proof | title = Derivation of canonical partition function (classical, discrete)
| proof =
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">
\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 constant ([[conservation of energy]]): <math display="block">
\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) + \lambda_1 \left( 1 - \sum_i \rho_i \right) + \lambda_2 \left( U - \sum_i \rho_i E_i \right) .</math>
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} - \lambda_1 - \lambda_2 E_i .</math>
Isolating for <math> \rho_i </math> yields
<math display="block">\rho_i = \exp \left( \frac{-k_\text{B} - \lambda_1 - \lambda_2 E_i}{k_\text{B}} \right) .</math>
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 number defined as the canonical ensemble partition function''':
<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 = k_\text{B} \ln(Z) - k_\text{B} </math>.
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} = \lambda_2 \equiv \frac{1}{T} .</math>
(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{
one can show that the additional contributions to this derivative cancel each other.)
Thus the canonical partition function <math> Z </math> becomes
<math display="block">Z \equiv \sum_i e^{-\beta E_i} ,</math>
where <math> \beta \equiv 1/(k_\text{B} T) </math> is defined as the [[thermodynamic beta]]. Finally, the probability distribution <math> \rho_i </math> and entropy <math> S </math> are respectively
<math display="block">\begin{align}
\rho_i & = \frac{1}{Z} e^{-\beta E_i} , \\
S & = \frac{U}{T} + k_\text{B} \ln Z .
\end{align}</math>
}}
==== 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)} \, d^3 q \, d^3 p, </math>
where
* <math> h </math> is the [[Planck constant]];
* <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
* <math> H(q, p) </math> is the [[Hamiltonian mechanics|Hamiltonian]] of the system;
* <math> q </math> is the [[Canonical coordinates|canonical position]];
* <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 constant]]).
For generalized cases, the partition function of <math> N </math> particles in <math> d </math>-dimensions is given by
<math display="block"> Z = \frac{1}{h^{Nd}} \int \prod_{i=1}^{N} e^{-\beta \mathcal{H}(\textbf{q}_i, \textbf{p}_i)} \, d^d \textbf{q}_i \, d^d \textbf{p}_i, </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]];
* <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
* <math> i </math> is the index for the particles of the system;
* <math> H </math> is the [[Hamiltonian mechanics|Hamiltonian]] of a respective particle;
* <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> d^3 </math> is shorthand notation to indicate that <math> q_i </math> and <math> p_i </math> are vectors in three-dimensional space.
* <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 constant).
==== 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:
<math display="block"> Z = \operatorname{tr} ( e^{-\beta \hat{H}} ), </math>
where:
* <math> \operatorname{tr} ( \circ ) </math> is the [[trace (linear algebra)|trace]] of a matrix;
* <math> \beta </math> is the [[thermodynamic beta]], defined as <math> \tfrac{1}{k_\text{B} T} </math>;
* <math> \hat{H} </math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]].
==== 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 \right\vert e^{-\beta \hat{H}} \left\vert q, p \right\rangle \, dq \, dp, </math>
where:
* <math> h </math> is the [[Planck constant]];
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 \, e^{-\beta E_j},</math>
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>''.
The above treatment applies to ''quantum'' [[statistical mechanics]], where a physical system inside a [[Particle in a box|finite-sized box]] will typically have a discrete set of energy eigenstates, which we can use as the states ''s'' above. In quantum mechanics, the partition function can be more formally written as a trace over the [[mathematical formulation of quantum mechanics|state space]] (which is independent of the choice of [[basis (linear algebra)|basis]]):
<math display="block">Z = \operatorname{tr} ( e^{-\beta \hat{H}} ),</math>
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}
= \int \langle x,p| e^{-\beta\hat{H}} |x, p\rangle \frac{dx \,dp}{h}.
</math>
A coherent state is an approximate eigenstate of both operators <math> \hat{x} </math> and <math> \hat{p} </math>, hence also of the Hamiltonian {{math|''Ĥ''}}, with errors of the size of the uncertainties. If {{math|Δ''x''}} and {{math|Δ''p''}} can be regarded as zero, the action of {{math|''Ĥ''}} reduces to multiplication by the classical Hamiltonian, and {{math|''Z''}} reduces to the classical configuration integral.
=== Connection to probability theory ===
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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 proportional to the number of microstates of the total [[Closed system (thermodynamics)|closed system]] (''S'', ''B'') in which ''S'' is in microstate ''i'' with energy ''E<sub>i</sub>''. Equivalently, ''p<sub>i</sub>'' will be proportional to the number of microstates of the heat bath ''B'' with energy {{nowrap|''E'' − ''E<sub>i</sub>''}}. We then normalize this by dividing by the total number of microstates in which the constraints we have imposed on the entire system; both S and the heat bath; hold. In this case the only constraint is that the total energy of both systems is ''E'', so:
<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]
&\approx -\frac{\partial\big(k \ln \Omega_B(E)\big)}{\partial E} E_i + k \ln\Omega_B(E) - k \ln \Omega_{(S,B)}(E)
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&\approx -\frac{\partial S_B}{\partial E} E_i + k \ln \frac{\Omega_B(E)}{\Omega_{(S,B)}(E)} \\[5pt]
&\approx -\frac{E_i}{T} + k \ln \frac{\Omega_B(E)}{\Omega_{(S,B)}(E)}
\end{align}</math>
Thus
<math display="block">p_i \propto e^{-E_i/(kT)} = e^{-\beta E_i}.</math>
Since the total probability to find the system in ''some'' microstate (the sum of all ''p<sub>i</sub>'') must be equal to 1, we know that the constant of proportionality must be the [[Normalizing constant|normalization constant]],
<math display="block"> Z = \sum_i e^{-\beta E_i} = \frac{\Omega_{(S,B)}(E)}{\Omega_B(E)}.</math>
=== Calculating the thermodynamic total energy ===
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}
&= \frac{1}{Z} \sum_s E_s e^{- \beta E_s} \\[1ex] \end{align}
</math>
or, equivalently,
<math display="block">\langle E\rangle = k_\text{B} T^2 \frac{\partial \ln Z}{\partial T}.</math>
Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner
<math display="block">E_s = E_s^{(0)} + \lambda A_s \qquad \text{for all}\; s </math>
then the expected value of ''A'' is
<math display="block">\langle A\rangle = \sum_s A_s P_s = -\frac{1}{\beta}
\frac{\partial}{\partial\lambda} \ln Z(\beta,\lambda).</math>
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As we have already seen, the thermodynamic energy is
<math display="block">\langle E \rangle = - \frac{\partial \ln Z}{\partial \beta}.</math>
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
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
In the special case of [[entropy]], entropy is given by
<math display="block">S \equiv -k_\text{B}\sum_s P_s \ln P_s = k_\text{B} (\ln Z + \beta \langle E\rangle) = \frac{\partial}{\partial T} (k_\text{B} T \ln Z) = -\frac{\partial A}{\partial T}</math>
where ''A'' is the [[Helmholtz free energy]] defined as {{math|1=''A'' = ''U'' − ''TS''}}, where {{math|1=''U'' = {{langle}}''E''{{rangle}}}} is the total energy and ''S'' is the [[entropy]], so that
<math display="block">A = \langle E\rangle -TS= - k_\text{B} T \ln Z.</math>
Furthermore, the heat capacity can be expressed as
<math display="block">C_\text{v} = T \frac{\partial S}{\partial T} = -T \frac{\partial^2 A}{\partial T^2}.</math>
=== 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'' [[factorial]]):
<math display="block">Z = \frac{\zeta^N}{N!}.</math>
This is to ensure that we do not "over-count" the number of microstates. While this may seem like a strange requirement, it is actually necessary to preserve the existence of a thermodynamic limit for such systems. This is known as the [[Gibbs paradox]].
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The partition function can be related to thermodynamic properties because it has a very important statistical meaning. The probability ''P<sub>s</sub>'' that the system occupies microstate ''s'' is
<math display="block">P_s = \frac{1}{Z} e^{- \beta E_s}. </math>
Thus, as shown above, the partition function plays the role of a normalizing constant (note that it does ''not'' depend on ''s''), ensuring that the probabilities sum up to one:
<math display="block">\sum_s P_s = \frac{1}{Z} \sum_s e^{- \beta E_s} = \frac{1}{Z} Z
= 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 |first=Kerson |title=Statistical Mechanics |publisher=John Wiley & Sons |___location=New York |year=1967 |isbn=0-471-81518-7 }}
* {{cite book |first=A. |last=Isihara |title=Statistical Physics |publisher=Academic Press |___location=New York |year=1971 |isbn=0-12-374650-7 }}
* {{cite web |last=Kelly |first=James J. |url=https://www.physics.umd.edu/courses/Phys603/kelly/Notes/IdealQuantumGases.pdf |title=Ideal Quantum Gases |work=Lecture notes |year=2002 }}
* {{cite book |first=L. D. |last=Landau |first2=E. M. |last2=Lifshitz |title=Statistical Physics |edition=3rd |others=Part 1 |publisher=Butterworth-Heinemann |___location=Oxford |year=1996 |isbn=0-08-023039-3 }}
* {{cite web |last=Vu-Quoc |first=L. |url=http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 |title=Configuration integral (statistical mechanics) |year=2008 |archive-url=https://web.archive.org/web/20120428193950/http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 |archive-date=April 28, 2012 |url-status=dead }}
{{refend}}
{{Statistical mechanics topics}}
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