Statistical Mechanics – Partition Function

Previously, we derived the state probability distribution for the canonical ensemble. The quantity p_\alpha, given by

\displaystyle p_\alpha=\frac{e^{-\beta E_\alpha}}{\sum_\alpha e^{-\beta E_\alpha}}

was the probability that a closed system occupies the state \alpha with associated energy E_\alpha. The variable \beta is as yet unendowed with physical meaning, but we can tell that, since it appears in an exponent with an energy term, it must have units of reciprocal energy.

For sake of ease, we give the sum on the denominator of p_\alpha its own special letter:

\displaystyle Z=\sum_\alpha e^{-\beta E_\alpha}

This sum is given the stately name of the partition function. Unwieldy as it appears, it turns out that any thermodynamic property of the system you’d care to ask out can be written in terms of the partition function.

Note that to evaluate Z we need to know how the energy levels \{E_\alpha\} of the system are discretised. This requires that we understand the system’s microphysics. So for now we’ll have to talk in general terms. But if we have the set \{E_\alpha\}, the partition function is the means by which we translate this microphysical knowledge to macrophysical knowledge.

 

As an example, consider the mean internal energy U of the system. It is related to the set of probabilities \{p_\alpha\} by

\displaystyle U=\sum_\alpha p_\alpha E_\alpha

Given the expression for p_\alpha above, we can rewrite this as

\displaystyle U=\sum_\alpha\frac{1}{Z}E_\alpha e^{-\beta E_\alpha}

The reciprocal of Z can be taken out of the sum:

\displaystyle U=\frac{1}{Z}\sum_\alpha E_\alpha e^{-\beta E_\alpha}

By inspection we see this is equal to

\displaystyle U=-\frac{1}{Z}\frac{\partial}{\partial \beta}\sum_\alpha e^{-\beta E_\alpha}

\displaystyle U=-\frac{1}{Z}\frac{\partial Z}{\partial \beta}

This is in fact equivalent to

\displaystyle U=-\frac{\partial \ln Z}{\partial \beta}

applying the chain rule ‘in reverse’. So if we know the partition function, we know the internal energy of the system!

 

Here’s a further example. Recall the expression for the Gibbs entropy:

\displaystyle S_G=-\sum_\alpha p_\alpha\ln p_\alpha

Again, we use the expression for p_\alpha:

\displaystyle S_G=-\sum_\alpha p_\alpha\ln\Bigg(\frac{e^{-\beta E_\alpha}}{Z}\Bigg)

\displaystyle S_G=\beta\sum_\alpha p_\alpha E_\alpha+\ln Z\sum_\alpha p_\alpha

S_G=\beta U+\ln Z

\displaystyle S_G=-\beta\frac{\partial \ln Z}{\partial \beta}+\ln Z

So if we know the partition function, we know the Gibbs entropy of the system.

 

This stuff is still pretty abstract. In a few posts’ time, we’ll actually look at a toy system to see how everything we’ve done so far comes together. But there’s a little more work to do yet.

In the next post, we’ll consider small changes to the Gibbs entropy, and in doing so, construct the first law of thermodynamics for a closed system.

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