Partition function (statistical mechanics): Difference between revisions

Revision as of 14:57, 13 September 2024

Latest revision as of 08:22, 7 August 2025

(14 intermediate revisions by 6 users not shown)

Line 36:

</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>

Line 46:

<math display="block">\begin{align}

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 \biggleft[ \delta {\Bigleft( - k_\text{B} \rho_i \ln \rho_i \Bigright) } - \delta {\Bigleft( \lambda_1 \rho_i \Bigright) } - \delta {\Bigleft( \lambda_2 E_i \rho_i \Bigright)} \biggright] \\

&= \sum_i \left[ \frac{\partial}{\partial \rho_i } \Bigleft( - k_\text{B} \rho_i \ln \rho_i \Bigright) \, \delta ( \rho_i ) - \frac{\partial}{\partial \rho_i } \Bigleft( \lambda_1 \rho_i \Bigright) \, \delta ( \rho_i ) - \frac{\partial}{\partial \rho_i } \Bigleft( \lambda_2 E_i \rho_i \Bigright) \, \delta ( \rho_i ) \right] \\[1ex]

&= \sum_i \biggleft[ -k_\text{B} \ln \rho_i - k_\text{B} - \lambda_1 - \lambda_2 E_i \biggright] \, \delta ( \rho_i ) .

\end{align}</math>

Line 63:

&= \exp \left( \frac{-k_\text{B} - \lambda_1}{k_\text{B}} \right) Z ,

\end{align}</math>

where '''<math> Z </math> is a constant 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>

Line 81:

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{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

Line 94 ⟶ 98:

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)} \, \mathrm{d}^3 q \, \mathrm{d}^3 p, </math>

where

* <math> h </math> is the [[Planck constant]];

Line 103 ⟶ 107:

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 noninteractingnon-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) \; \mathrm{d}^3 q_1 \cdots \mathrm{d}^3 q_N \, \mathrm{d}^3 p_1 \cdots \mathrm{d}^3 p_N = \frac{Z_{\text{single}}^N}{N!}</math>

where

* <math> h </math> is the [[Planck constant]];

Line 115 ⟶ 123:

* <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> \mathrm{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.

Line 134 ⟶ 142:

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 \, \mathrm{d} qdq \, \mathrm{d} pdp, </math>

where:

* <math> h </math> is the [[Planck constant]];

Line 143 ⟶ 151:

In systems with multiple [[quantum states]] ''s'' sharing the same energy ''E_s'', 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 \cdot, e^{-\beta E_j},</math>

where ''g_j'' is the degeneracy factor, or number of quantum states ''s'' that have the same energy level defined by ''E_j'' = ''E_s''.

Line 163 ⟶ 171:

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_i'' denote the [[probability]] that the system ''S'' is in a particular [[Microstate (statistical mechanics)|microstate]], ''i'', with energy ''E_i''. 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_i'' will be inversely 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_i''. Equivalently, ''p_i'' will be proportional to the number of microstates of the heat bath ''B'' with energy {{nowrap|''E'' − ''E_i''}}. 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>

Line 184 ⟶ 192:

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]

e^{- \beta E_s} &= - \frac{1}{Z} \frac{\partial}{\partial \beta} Z(\beta, E_1, E_2, \dots) \\[1ex]

Z(\beta, E_1, E_2, \cdots) &= - \frac{\partial \ln Z}{\partial \beta}

\end{align}

</math>

or, equivalently,

Line 207 ⟶ 218:

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

E\rangle)^2 \rangle = \left\langle E^2 \right\rangle - {\left\langle E \right\rangle}^2
= \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:

Line 217 ⟶ 229:

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

Line 231 ⟶ 243:

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''₁, ''ζ''₂, ..., ''ζ''_N, 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''₁ = ''ζ''₂ = ... = ''ζ'', in which case <math display="block">Z = \zeta^N.</math>

Line 259 ⟶ 271:

The grand canonical partition function, denoted by <math>\mathcal{Z}</math>, is the following sum over [[microstate (statistical mechanics)|microstates]]

: <math display="block"> \mathcal{Z}(\mu, V, T) = \sum_{i}sum_i \exp\left(\frac{N_i\mu - E_i}{k_B T} \right). </math>

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

: <math display="block"> -k_\text{B} T \ln \mathcal{Z} = \Phi_{\rm G} = \langle E \rangle - TS - \mu \langle N\rangle. </math>

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>:

: <math display="block"> p_i = \frac{1}{\mathcal Z} \exp\left(\frac{N_i \mu - E_i}{k_B T}\right).</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>

: <math display="block"> \mathcal{Z}(z, V, T) = \sum_{N_i} z^{N_i} Z(N_i, V, T), </math>

where <math>z \equiv \exp(\mu/k_\text{B} T)</math> is known as the absolute [[activity (chemistry)|activity]] (or [[fugacity]]) and <math>Z(N_i, V, T)</math> is the canonical partition function.

Line 281 ⟶ 293:

== 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 }}

Line 286 ⟶ 299:

* {{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}}