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Partition function (statistical mechanics)

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Function in thermodynamics and statistical physics

Statistical mechanics

Thermodynamics Kinetic theory
Particle statistics Spin–statistics theorem Indistinguishable particles Maxwell–Boltzmann Bose–Einstein Fermi–Dirac Parastatistics Anyonic statistics Braid statistics
Thermodynamic ensembles NVEMicrocanonical NVTCanonical µVTGrand canonical NPHIsoenthalpic–isobaric NPTIsothermal–isobaric
Models Debye Einstein Ising Potts
Potentials Internal energy Enthalpy Helmholtz free energy Gibbs free energy Grand potential / Landau free energy
Scientists Maxwell Boltzmann Helmholtz Bose Gibbs Einstein Dirac Ehrenfest von Neumann Tolman Debye Fermi Synge Ising Landau
v t e

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.

Inphysics, apartition function describes thestatistical properties of a system inthermodynamic equilibrium.^{[citation needed]} Partition functions arefunctions of the thermodynamicstate variables, such as thetemperature andvolume. Most of the aggregatethermodynamic variables of the system, such as thetotal energy,free energy,entropy, andpressure, can be expressed in terms of the partition function or itsderivatives. The partition function is dimensionless.

Each partition function is constructed to represent a particularstatistical ensemble (which, in turn, corresponds to a particularfree energy). The most common statistical ensembles have named partition functions. Thecanonical partition function applies to acanonical ensemble, in which the system is allowed to exchangeheat with theenvironment at fixed temperature, volume, andnumber of particles. Thegrand canonical partition function applies to agrand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, andchemical potential. Other types of partition functions can be defined for different circumstances; seepartition function (mathematics) for generalizations. The partition function has many physical meanings, as discussed inMeaning and significance.

Canonical partition function

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Definition

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Initially, let us assume that a thermodynamically large system is inthermal contact with the environment, with a temperatureT, 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 acanonical ensemble. The appropriatemathematical expression for the canonical partition function depends on thedegrees of freedom of the system, whether the context isclassical mechanics orquantum mechanics, and whether the spectrum of states isdiscrete orcontinuous.^{[citation needed]}

Classical discrete system

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For a canonical ensemble that is classical and discrete, the canonical partition function is defined as $Z=\sum _{i}e^{-\beta E_{i}},$ where

$i {\displaystyle i}$ is the index for themicrostates of the system;
$e {\displaystyle e}$ isEuler's number;
$\beta$ is thethermodynamic beta, defined as ${\tfrac {1}{k_{\text{B}}T}}$ where $k_{\text{B}}$ is theBoltzmann constant;
$E_{i}$ is the total energy of the system in the respectivemicrostate.

Theexponential factor $e^{-\beta E_{i}}$ is otherwise known as theBoltzmann factor.

Derivation of canonical partition function (classical, discrete)

There are multiple approaches to deriving the partition function. The following derivation follows the more powerful and generalinformation-theoretic Jaynesian maximum entropy approach.

According to thesecond law of thermodynamics, a system assumes a configuration ofmaximum entropy atthermodynamic equilibrium. We seek a probability distribution of states $\rho _{i}$ that maximizes the discreteGibbs entropy $S=-k_{\text{B}}\sum _{i}\rho _{i}\ln \rho _{i}$ subject to two physical constraints:

The probabilities of all states add to unity (second axiom of probability): $\sum _{i}\rho _{i}=1.$
In thecanonical ensemble, the system is inthermal equilibrium, so the average energy does not change over time; in other words, the average energy is constant (conservation of energy): $\langle E\rangle =\sum _{i}\rho _{i}E_{i}\equiv U.$

Applyingvariational calculus with constraints (analogous in some sense to the method ofLagrange multipliers), we write the Lagrangian (or Lagrange function) ${\mathcal {L}}$ as ${\mathcal {L}}=\left(-k_{\text{B}}\sum _{i}\rho _{i}\ln \rho _{i}\right)+\lambda _{1}\left(1-\sum _{i}\rho _{i}\right)+\lambda _{2}\left(U-\sum _{i}\rho _{i}E_{i}\right).$

Varying and extremizing ${\mathcal {L}}$ with respect to $\rho _{i}$ leads to ${\begin{aligned}0&\equiv \delta {\mathcal {L}}\\&=\delta {\left(-\sum _{i}k_{\text{B}}\rho _{i}\ln \rho _{i}\right)}+\delta {\left(\lambda _{1}-\sum _{i}\lambda _{1}\rho _{i}\right)}+\delta {\left(\lambda _{2}U-\sum _{i}\lambda _{2}\rho _{i}E_{i}\right)}\\[1ex]&=\sum _{i}\left[\delta {\left(-k_{\text{B}}\rho _{i}\ln \rho _{i}\right)}-\delta {\left(\lambda _{1}\rho _{i}\right)}-\delta {\left(\lambda _{2}E_{i}\rho _{i}\right)}\right]\\&=\sum _{i}\left[{\frac {\partial }{\partial \rho _{i}}}\left(-k_{\text{B}}\rho _{i}\ln \rho _{i}\right)\delta \rho _{i}-{\frac {\partial }{\partial \rho _{i}}}\left(\lambda _{1}\rho _{i}\right)\delta \rho _{i}-{\frac {\partial }{\partial \rho _{i}}}\left(\lambda _{2}E_{i}\rho _{i}\right)\delta \rho _{i}\right]\\[1ex]&=\sum _{i}\left[-k_{\text{B}}\ln \rho _{i}-k_{\text{B}}-\lambda _{1}-\lambda _{2}E_{i}\right]\delta \rho _{i}.\end{aligned}}$

Since this equation should hold for any variation $\delta (\rho _{i})$ , it implies that $0\equiv -k_{\text{B}}\ln \rho _{i}-k_{\text{B}}-\lambda _{1}-\lambda _{2}E_{i}.$

Isolating for $\rho _{i}$ yields $\rho _{i}=\exp \left({\frac {-k_{\text{B}}-\lambda _{1}-\lambda _{2}E_{i}}{k_{\text{B}}}}\right).$

To obtain $\lambda _{1}$ , one substitutes the probability into the first constraint: ${\begin{aligned}1&=\sum _{i}\rho _{i}\\&=\exp \left({\frac {-k_{\text{B}}-\lambda _{1}}{k_{\text{B}}}}\right)Z,\end{aligned}}$ where $Z {\displaystyle Z}$ is a number defined as the canonical ensemble partition function: $Z\equiv \sum _{i}\exp \left(-{\frac {\lambda _{2}}{k_{\text{B}}}}E_{i}\right).$

Isolating for $\lambda _{1}$ yields $\lambda _{1}=k_{\text{B}}\ln(Z)-k_{\text{B}}$ .

Rewriting $\rho _{i}$ in terms of $Z {\displaystyle Z}$ gives $\rho _{i}={\frac {1}{Z}}\exp \left(-{\frac {\lambda _{2}}{k_{\text{B}}}}E_{i}\right).$

Rewriting $S {\displaystyle S}$ in terms of $Z {\displaystyle Z}$ gives ${\begin{aligned}S&=-k_{\text{B}}\sum _{i}\rho _{i}\ln \rho _{i}\\&=-k_{\text{B}}\sum _{i}\rho _{i}\left(-{\frac {\lambda _{2}}{k_{\text{B}}}}E_{i}-\ln(Z)\right)\\&=\lambda _{2}\sum _{i}\rho _{i}E_{i}+k_{\text{B}}\ln(Z)\sum _{i}\rho _{i}\\&=\lambda _{2}U+k_{\text{B}}\ln(Z).\end{aligned}}$

To obtain $\lambda _{2}$ , we differentiate $S {\displaystyle S}$ with respect to the average energy $U {\displaystyle U}$ and apply thefirst law of thermodynamics, $dU=TdS-PdV$ : ${\frac {dS}{dU}}=\lambda _{2}\equiv {\frac {1}{T}}.$

(Note that $\lambda _{2}$ and $Z {\displaystyle Z}$ vary with $U {\displaystyle U}$ as well; however, using the chain rule and ${\frac {d}{d\lambda _{2}}}\ln(Z)=-{\frac {1}{k_{\text{B}}}}\sum _{i}\rho _{i}E_{i}=-{\frac {U}{k_{\text{B}}}},$ one can show that the additional contributions to this derivative cancel each other.)

Thus the canonical partition function $Z {\displaystyle Z}$ becomes $Z\equiv \sum _{i}e^{-\beta E_{i}},$ where $\beta \equiv 1/(k_{\text{B}}T)$ is defined as thethermodynamic beta. Finally, the probability distribution $\rho _{i}$ and entropy $S {\displaystyle S}$ are respectively ${\begin{aligned}\rho _{i}&={\frac {1}{Z}}e^{-\beta E_{i}},\\S&={\frac {U}{T}}+k_{\text{B}}\ln Z.\end{aligned}}$

Classical continuous system

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Inclassical mechanics, theposition andmomentum variables of a particle can vary continuously, so the set of microstates is actuallyuncountable. Inclassical statistical mechanics, it is rather inaccurate to express the partition function as asum of discrete terms. In this case we must describe the partition function using anintegral rather than a sum. For a canonical ensemble that is classical and continuous, the canonical partition function is defined as $Z={\frac {1}{h^{3}}}\int e^{-\beta H(q,p)}\,d^{3}q\,d^{3}p,$ where

$h {\displaystyle h}$ is thePlanck constant;
$\beta$ is thethermodynamic beta, defined as ${\tfrac {1}{k_{\text{B}}T}}$ ;
$H(q,p)$ is theHamiltonian of the system;
$q {\displaystyle q}$ is thecanonical position;
$p {\displaystyle p}$ is thecanonical momentum.

To make it into a dimensionless quantity, we must divide it byh, which is some quantity with units ofaction (usually taken to be thePlanck constant).

For generalized cases, the partition function of $N {\displaystyle N}$ particles in $d {\displaystyle d}$ -dimensions is given by

$Z={\frac {1}{h^{Nd}}}\int \prod _{i=1}^{N}e^{-\beta {\mathcal {H}}({\textbf {q}}_{i},{\textbf {p}}_{i})}\,d^{d}{\textbf {q}}_{i}\,d^{d}{\textbf {p}}_{i},$

Classical continuous system (multiple identical particles)

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For a gas of $N {\displaystyle N}$ identical classical non-interacting particles in three dimensions, the partition function is $Z={\frac {1}{N!h^{3N}}}\int \,\exp \left(-\beta \sum _{i=1}^{N}H({\textbf {q}}_{i},{\textbf {p}}_{i})\right)\;d^{3}q_{1}\cdots d^{3}q_{N}\,d^{3}p_{1}\cdots d^{3}p_{N}={\frac {Z_{\text{single}}^{N}}{N!}}$ where

$h {\displaystyle h}$ is thePlanck constant;
$\beta$ is thethermodynamic beta, defined as ${\tfrac {1}{k_{\text{B}}T}}$ ;
$i {\displaystyle i}$ is the index for the particles of the system;
$H {\displaystyle H}$ is theHamiltonian of a respective particle;
$q_{i}$ is thecanonical position of the respective particle;
$p_{i}$ is thecanonical momentum of the respective particle;
$d^{3}$ is shorthand notation to indicate that $q_{i}$ and $p_{i}$ are vectors in three-dimensional space.
$Z_{\text{single}}$ is the classical continuous partition function of a single particle as given in the previous section.

The reason for thefactorial factorN! is discussedbelow. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is notdimensionless. As stated in the previous section, to make it into a dimensionless quantity, we must divide it byh^3N (whereh is usually taken to be the Planck constant).

Quantum mechanical discrete system

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For a canonical ensemble that is quantum mechanical and discrete, the canonical partition function is defined as thetrace of the Boltzmann factor: $Z=\operatorname {tr} (e^{-\beta {\hat {H}}}),$ where:

$\operatorname {tr} (\circ )$ is thetrace of a matrix;
$\beta$ is thethermodynamic beta, defined as ${\tfrac {1}{k_{\text{B}}T}}$ ;
${\hat {H}}$ is theHamiltonian operator.

Thedimension of $e^{-\beta {\hat {H}}}$ is the number ofenergy eigenstates of the system.

Quantum mechanical continuous system

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For a canonical ensemble that is quantum mechanical and continuous, the canonical partition function is defined as $Z={\frac {1}{h}}\int \left\langle q,p\right\vert e^{-\beta {\hat {H}}}\left\vert q,p\right\rangle \,dq\,dp,$ where:

$h {\displaystyle h}$ is thePlanck constant;
$\beta$ is thethermodynamic beta, defined as ${\tfrac {1}{k_{\text{B}}T}}$ ;
${\hat {H}}$ is theHamiltonian operator;
$q {\displaystyle q}$ is thecanonical position;
$p {\displaystyle p}$ is thecanonical momentum.

In systems with multiplequantum statess sharing the same energyE_s, it is said that theenergy levels of the system aredegenerate. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed byj) as follows: $Z=\sum _{j}g_{j}\,e^{-\beta E_{j}},$ whereg_j is the degeneracy factor, or number of quantum statess that have the same energy level defined byE_j =E_s.

The above treatment applies toquantumstatistical mechanics, where a physical system inside afinite-sized box will typically have a discrete set of energy eigenstates, which we can use as the statess above. In quantum mechanics, the partition function can be more formally written as a trace over thestate space (which is independent of the choice ofbasis): $Z=\operatorname {tr} (e^{-\beta {\hat {H}}}),$ whereĤ is thequantum Hamiltonian operator. The exponential of an operator can be defined using theexponential power series.

The classical form ofZ is recovered when the trace is expressed in terms ofcoherent states^[1] and when quantum-mechanicaluncertainties in the position and momentum of a particle are regarded as negligible. Formally, usingbra–ket notation, one inserts under the trace for each degree of freedom the identity: ${\boldsymbol {1}}=\int |x,p\rangle \langle x,p|{\frac {dx\,dp}{h}},$ where|x,p⟩ is anormalised Gaussian wavepacket centered at positionx and momentump. Thus $Z=\int \operatorname {tr} \left(e^{-\beta {\hat {H}}}|x,p\rangle \langle x,p|\right){\frac {dx\,dp}{h}}=\int \langle x,p|e^{-\beta {\hat {H}}}|x,p\rangle {\frac {dx\,dp}{h}}.$ A coherent state is an approximate eigenstate of both operators ${\hat {x}}$ and ${\hat {p}}$ , hence also of the HamiltonianĤ, with errors of the size of the uncertainties. IfΔx andΔp can be regarded as zero, the action ofĤ reduces to multiplication by the classical Hamiltonian, andZ 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 systemS embedded into aheat bathB. Let the totalenergy of both systems beE. Letp_i denote theprobability that the systemS is in a particularmicrostate,i, with energyE_i. According to thefundamental postulate of statistical mechanics (which states that all attainable microstates of a system are equally probable), the probabilityp_i will be inversely proportional to the number of microstates of the totalclosed system (S,B) in whichS is in microstatei with energyE_i. Equivalently,p_i will be proportional to the number of microstates of the heat bathB with energyE −E_i: $p_{i}={\frac {\Omega _{B}(E-E_{i})}{\Omega _{(S,B)}(E)}}.$

Assuming that the heat bath's internal energy is much larger than the energy ofS (E ≫E_i), we canTaylor-expand $\Omega _{B}$ to first order inE_i and use the thermodynamic relation $\partial S_{B}/\partial E=1/T$ , where here $S_{B}$ , $T {\displaystyle T}$ are the entropy and temperature of the bath respectively: ${\begin{aligned}k\ln p_{i}&=k\ln \Omega _{B}(E-E_{i})-k\ln \Omega _{(S,B)}(E)\\[5pt]&\approx -{\frac {\partial {\big (}k\ln \Omega _{B}(E){\big )}}{\partial E}}E_{i}+k\ln \Omega _{B}(E)-k\ln \Omega _{(S,B)}(E)\\[5pt]&\approx -{\frac {\partial S_{B}}{\partial E}}E_{i}+k\ln {\frac {\Omega _{B}(E)}{\Omega _{(S,B)}(E)}}\\[5pt]&\approx -{\frac {E_{i}}{T}}+k\ln {\frac {\Omega _{B}(E)}{\Omega _{(S,B)}(E)}}\end{aligned}}$

Thus $p_{i}\propto e^{-E_{i}/(kT)}=e^{-\beta E_{i}}.$

Since the total probability to find the system insome microstate (the sum of allp_i) must be equal to 1, we know that the constant of proportionality must be thenormalization constant, and so, we can define the partition function to be this constant: $Z=\sum _{i}e^{-\beta E_{i}}={\frac {\Omega _{(S,B)}(E)}{\Omega _{B}(E)}}.$

Calculating the thermodynamic total energy

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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 theexpected value, orensemble average for the energy, which is the sum of the microstate energies weighted by their probabilities: ${\begin{aligned}\langle E\rangle =\sum _{s}E_{s}P_{s}&={\frac {1}{Z}}\sum _{s}E_{s}e^{-\beta E_{s}}\\[1ex]&=-{\frac {1}{Z}}{\frac {\partial }{\partial \beta }}Z(\beta ,E_{1},E_{2},\dots )\\[1ex]&=-{\frac {\partial \ln Z}{\partial \beta }}\end{aligned}}$ or, equivalently, $\langle E\rangle =k_{\text{B}}T^{2}{\frac {\partial \ln Z}{\partial T}}.$

Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner $E_{s}=E_{s}^{(0)}+\lambda A_{s}\qquad {\text{for all}}\;s$ then the expected value ofA is $\langle A\rangle =\sum _{s}A_{s}P_{s}=-{\frac {1}{\beta }}{\frac {\partial }{\partial \lambda }}\ln Z(\beta ,\lambda ).$

This provides us with a method for calculating the expected values of many microscopic quantities. We add the quantity artificially to the microstate energies (or, in the language of quantum mechanics, to the Hamiltonian), calculate the new partition function and expected value, and then setλ to zero in the final expression. This is analogous to thesource field method used in thepath integral formulation ofquantum field theory.^{[citation needed]}

Relation to thermodynamic variables

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In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system. These results can be derived using the method of the previous section and the various thermodynamic relations.

As we have already seen, the thermodynamic energy is $\langle E\rangle =-{\frac {\partial \ln Z}{\partial \beta }}.$

Thevariance in the energy (or "energy fluctuation") is $\left\langle (\Delta E)^{2}\right\rangle \equiv \left\langle (E-\langle E\rangle )^{2}\right\rangle =\left\langle E^{2}\right\rangle -{\left\langle E\right\rangle }^{2}={\frac {\partial ^{2}\ln Z}{\partial \beta ^{2}}}.$

Theheat capacity is $C_{v}={\frac {\partial \langle E\rangle }{\partial T}}={\frac {1}{k_{\text{B}}T^{2}}}\left\langle (\Delta E)^{2}\right\rangle .$

In general, consider theextensive variableX andintensive variableY whereX andY form a pair ofconjugate variables. In ensembles whereY is fixed (andX is allowed to fluctuate), then the average value ofX will be: $\langle X\rangle =\pm {\frac {\partial \ln Z}{\partial \beta Y}}.$

The sign will depend on the specific definitions of the variablesX andY. An example would beX = volume andY = pressure. Additionally, the variance inX will be $\left\langle (\Delta X)^{2}\right\rangle \equiv \left\langle (X-\langle X\rangle )^{2}\right\rangle ={\frac {\partial \langle X\rangle }{\partial \beta Y}}={\frac {\partial ^{2}\ln Z}{\partial (\beta Y)^{2}}}.$

In the special case ofentropy, entropy is given by $S\equiv -k_{\text{B}}\sum _{s}P_{s}\ln P_{s}=k_{\text{B}}(\ln Z+\beta \langle E\rangle )={\frac {\partial }{\partial T}}(k_{\text{B}}T\ln Z)=-{\frac {\partial A}{\partial T}}$ whereA is theHelmholtz free energy defined asA =U −TS, whereU = ⟨E⟩ is the total energy andS is theentropy, so that $A=\langle E\rangle -TS=-k_{\text{B}}T\ln Z.$

Furthermore, the heat capacity can be expressed as $C_{\text{v}}=T{\frac {\partial S}{\partial T}}=-T{\frac {\partial ^{2}A}{\partial T^{2}}}.$

Partition functions of subsystems

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Suppose a system is subdivided intoN 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 areζ₁,ζ₂, ...,ζ_N, then the partition function of the entire system is theproduct of the individual partition functions: $Z=\prod _{j=1}^{N}\zeta _{j}.$

If the sub-systems have the same physical properties, then their partition functions are equal,ζ₁ =ζ₂ = ... =ζ, in which case $Z=\zeta ^{N}.$

However, there is a well-known exception to this rule. If the sub-systems are actuallyidentical particles, in thequantum mechanical sense that they are impossible to distinguish even in principle, the total partition function must be divided by aN! (N factorial): $Z={\frac {\zeta ^{N}}{N!}}.$

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 theGibbs paradox.

Meaning and significance

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It may not be obvious why the partition function, as we have defined it above, is an important quantity. First, consider what goes into it. The partition function is a function of the temperatureT and the microstate energiesE₁,E₂,E₃, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles. This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system.

The partition function can be related to thermodynamic properties because it has a very important statistical meaning. The probabilityP_s that the system occupies microstates is $P_{s}={\frac {1}{Z}}e^{-\beta E_{s}}.$

Thus, as shown above, the partition function plays the role of a normalizing constant (note that it doesnot depend ons), ensuring that the probabilities sum up to one: $\sum _{s}P_{s}={\frac {1}{Z}}\sum _{s}e^{-\beta E_{s}}={\frac {1}{Z}}Z=1.$

This is the reason for callingZ 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 theisothermal-isobaric ensemble, thegeneralized Boltzmann distribution, divides up probabilities based on particle number, pressure, and temperature. The energy is replaced by the characteristic potential of that ensemble, theGibbs Free Energy. The letterZ stands for theGerman wordZustandssumme, "sum over states". The usefulness of the partition function stems from the fact that the macroscopicthermodynamic quantities of a system can be related to its microscopic details through the derivatives of its partition function. Finding the partition function is also equivalent to performing aLaplace transform of the density of states function from the energy domain to theβ domain, and theinverse Laplace transform of the partition function reclaims the state density function of energies.

Grand canonical partition function

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Main article:Grand canonical ensemble

We can define agrand canonical partition function for agrand canonical ensemble, which describes the statistics of a constant-volume system that can exchange both heat and particles with a reservoir. The reservoir has a constant temperatureT, and achemical potentialμ.

The grand canonical partition function, denoted by ${\mathcal {Z}}$ , is the following sum overmicrostates ${\mathcal {Z}}(\mu ,V,T)=\sum _{i}\exp \left({\frac {N_{i}\mu -E_{i}}{k_{B}T}}\right).$ Here, each microstate is labelled by $i {\displaystyle i}$ , and has total particle number $N_{i}$ and total energy $E_{i}$ . This partition function is closely related to thegrand potential, $\Phi _{\rm {G}}$ , by the relation $-k_{\text{B}}T\ln {\mathcal {Z}}=\Phi _{\rm {G}}=\langle E\rangle -TS-\mu \langle N\rangle .$ This can be contrasted to the canonical partition function above, which is related instead to theHelmholtz 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 $i {\displaystyle i}$ : $p_{i}={\frac {1}{\mathcal {Z}}}\exp \left({\frac {N_{i}\mu -E_{i}}{k_{B}T}}\right).$

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^[2] ${\mathcal {Z}}(z,V,T)=\sum _{N_{i}}z^{N_{i}}Z(N_{i},V,T),$ where $z\equiv \exp(\mu /k_{\text{B}}T)$ is known as the absoluteactivity (orfugacity) and $Z(N_{i},V,T)$ is the canonical partition function.

References

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^Klauder, John R.; Skagerstam, Bo-Sture (1985).Coherent States: Applications in Physics and Mathematical Physics. World Scientific. pp. 71–73.ISBN 978-9971-966-52-2.
^Baxter, Rodney J. (1982).Exactly solved models in statistical mechanics. Academic Press Inc.ISBN 9780120831807.

Huang, Kerson (1967).Statistical Mechanics. New York: John Wiley & Sons.ISBN 0-471-81518-7.
Isihara, A. (1971).Statistical Physics. New York: Academic Press.ISBN 0-12-374650-7.
Kelly, James J. (2002)."Ideal Quantum Gases"(PDF).Lecture notes.
Landau, L. D.; Lifshitz, E. M. (1996).Statistical Physics. Part 1 (3rd ed.). Oxford: Butterworth-Heinemann.ISBN 0-08-023039-3.
Vu-Quoc, L. (2008)."Configuration integral (statistical mechanics)". Archived fromthe original on April 28, 2012.

v t e Statistical mechanics
Theory	Principle of maximum entropy ergodic theory
Statistical thermodynamics	Ensembles partition functions equations of state thermodynamic potential: U H F G Maxwell relations
Models	Ferromagnetism models Ising Potts Heisenberg percolation Particles withforce field depletion force Lennard-Jones potential
Mathematical approaches	Boltzmann equation H-theorem Vlasov equation BBGKY hierarchy stochastic process mean-field theory andconformal field theory
Critical phenomena	Phase transition Critical exponents correlation length size scaling
Entropy	Boltzmann Shannon Tsallis Rényi von Neumann
Applications	Statistical field theory elementary particle superfluidity Condensed matter physics Complex system chaos information theory Boltzmann machine