Partition function (statistical mechanics): Difference between revisions

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</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>

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<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>

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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]];

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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) \; \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]];

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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> \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.

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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]];

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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''.

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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_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>

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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]

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,

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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

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:

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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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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>

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== 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 }}

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* {{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}}