5.2 Two examples of the Gibbs Distribution

Take-home message: There is an important distinction between systems in which energy levels can only have one particle in them, and those where they can have many.

Example 1: Sites which bind a single molecule

We first consider sites on a surface which can bind a single molecule only; the energy of the empty site is 0 and that of the occupied site is $\varepsilon_0$ (which can have either sign, but is negative for binding). This example is common in biology, where receptor molecules can be occupied or unoccupied.

The surface is in contact with a gas or a solution with chemical potential $\mu$ (the energy drop of the solution when it loses a molecule). What is the grand partition function, and the average occupancy of a site?

There are only two microstates here: unoccupied, with $N=0$ and $\varepsilon=0$, and occupied, with $N=1$ and $\varepsilon=\varepsilon_0$, so there are only two terms in the grand partition function:

$${\cal Z}=e^0+e^{(\mu-\varepsilon_0)\beta}=1+e^{(\mu-\varepsilon_0)\beta}$$

Then

$$\left\langle N \right\rangle =-\left({\partial(-k_{\scriptsc... ...\!\scriptstyle \beta}= {1\over e^{(\varepsilon_0-\mu)\beta}+1}$$

Below we plot the average occupancy as a function of $\varepsilon_0$, the energy of the level in question.

We see that $\left\langle N \right\rangle$ is always less than 1, as it must be. If a level lies above the chemical potential, $\varepsilon_0>\mu$ then it is less likely to be occupied, since it is energetically more favourable for the molecule to remain in solution. Conversely if $\varepsilon_0<\mu$ then it is more likely to be occupied, since that is the energetically favourable configuration. As always, it is the temperature which determines the likelihood of the less favourable configuration obtaining. At zero temperature, the distribution becomes a step function, with $\left\langle N \right\rangle =1$ if $\varepsilon_0<\mu$ and $\left\langle N \right\rangle =0$ if $\varepsilon_0>\mu$.

Example 2: sites which bind many molecules

This is less realistic, but we imagine a site which can have any number of molecules occupying it, with energy $\varepsilon_0$ per molecule. There are then infinitely many terms in the grand partition function which form a geometric series:

$$\begin{eqnarray*} {\cal Z}\!\!\!&=&\!\!\!e^0+e^{(\mu-\varepsilon_0)\beta}+e^{2(\... ...ht\rangle \!\!\!&=&\!\!\!{1\over e^{(\varepsilon_0-\mu)\beta}-1} \end{eqnarray*}$$

Below we plot the average occupancy as a function of $\varepsilon_0$, the energy of the level in question.

Unlike the first example, there is no limit to $\left\langle N \right\rangle$. Thus it doesn't make sense to consider states with $\varepsilon_0<\mu$, as their occupancy will be infinite. (The formula above for $\left\langle N \right\rangle$ is no longer valid in that case.) For $\varepsilon_0$ close to $\mu$ the occupancy will be high, and it falls off as $\varepsilon_0$ increases. The rapidity of the drop depends on temperature; for $T=0$ only a level with $\varepsilon_0=\mu$ would have non-zero occupancy.